Magnetohydrodynamics

S. I. Syrovatskii

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
1. Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
2. Propagation of small disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
3. Discontinuity surfaces and shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4. Some solutions of the equations of magnetohydrodynamics . . . . . . . . . . . . . 268
5. Amplification of the magnetic field. The hydromagnetic dynamo . . . . . . . . 275
6. Stability problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
7. Magnetohydrodynamic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Introduction

Magnetohydrodynamics studies the interaction of an electromagnetic field with a liquid or gaseous conductor, considered as a continuous medium. Its theoretical foundation consists of the classical equations of the electromagnetic field and the hydrodynamic equations of motion of a continuous medium. Until recently this range of questions remained outside the field of view of physics. The reason is that the phenomena characteristic of magnetohydrodynamics can be detected only in an extended liquid or gaseous medium possessing high electrical conductivity. As a rule, one encounters such a medium in astrophysics, but it is difficult to realize it under laboratory conditions.

The exceptions are mercury and molten metals. The study of the dynamics of mercury in a magnetic field74, 75 was in fact one of the first investigations in magnetohydrodynamics. These works were prompted by the idea of an electromagnetic mercury pump, but because of its insufficient efficiency they did not receive further development. Only in recent years, in connection with the development of the technology of nuclear reactors with a metallic coolant, has interest again arisen in electromagnetic pumps and measuring instruments based on magnetohydrodynamic effects123a, 135.

Some magnetohydrodynamic phenomena have long been considered in connection with theories of terrestrial and stellar magnetism56, 45. However, the actual emergence and rapid development of magnetohydrodynamics as an independent branch of physics is due entirely to a number of new problems in astrophysics, among which the problems of the motion of cosmic gas masses and the origin of cosmic rays are of primary importance.

The need to introduce into the study of a number of astrophysical problems the theory of the interaction of a conducting medium with an electromagnetic field was first clearly formulated by Alfvén5. He drew attention to the following well-known circumstances. First, interstellar gas, the atmospheres of stars, and the matter inside stars are highly ionized

and, consequently, are excellent electrical conductors. Secondly, many cosmic objects possess electric and, in particular, magnetic fields. These facts formed the basis for a whole series of studies on the dynamics of cosmic gaseous masses \(^{120,121,59}\), the origin of cosmic rays \(^{128,71}\) and of cosmic radio emission \(^{71,129}\), the origin of terrestrial and stellar magnetism \(^{56,59}\), the polarization of light from distant stars \(^{122}\), and others. At the present time, cosmic electromagnetic processes constitute an important branch of astrophysics. This has necessitated the intensive development of the theory of such processes.

In many practically important cases, cosmic gaseous masses, despite their extremely low density, may be regarded as a continuous medium. The point is that the linear dimensions of the phenomena considered in astrophysics, as a rule, far exceed the mean free path of the gas particles, which makes the hydrodynamic approach to the study of these phenomena fully justified. Thus an important class of large-scale processes in cosmic physics can be studied by the methods of magnetic hydrodynamics.

Recently, magnetohydrodynamic phenomena have also been observed in gas discharges \(^{16-19,7,14}\). The study of the behavior of plasma in a magnetic field is apparently the most promising method for modeling cosmic electromagnetic processes and for investigating magnetohydrodynamic phenomena under laboratory conditions.

Although magnetic hydrodynamics is a young, rapidly developing field of physics, a number of important results have already been obtained in it which make it possible to reveal the characteristic features of the circle of phenomena under consideration. The principal results are presented in the present review. With few exceptions, the numerous applications of magnetic hydrodynamics to specific problems of physics and astrophysics are not considered in the review for lack of space. Many of these applications are now the subject of special reviews. For the same reasons, the connection between magnetic hydrodynamics and the microscopic theory of ionized gas is not considered. The detailed bibliography at the end of the review makes it possible to supplement the basic information set forth below.

1. BASIC EQUATIONS

The equations of magnetic hydrodynamics are a combination of Maxwell’s equations for the electromagnetic field and the usual hydrodynamic equations describing the motion of a continuous medium—a liquid or a gas. The connection between these two groups of equations is due, on the one hand, to the appearance of an induction current when a conducting medium moves in a magnetic field. This current must be taken into account in Maxwell’s equations. On the other hand, the action of the magnetic field on the currents in the medium leads to an additional electromagnetic volume force, which must be taken into account in the hydrodynamic equations.

Thus the intensities of the electric and magnetic fields \(\mathbf{E}\) and \(\mathbf{H}\) obey the well-known equations

\[ \operatorname{rot}\mathbf{H}=\frac{4\pi}{c_0}\mathbf{j}+\frac{\varepsilon_0}{c_0}\frac{\partial \mathbf{E}}{\partial t}, \tag{1,1} \]

\[ \operatorname{rot}\mathbf{E}=-\frac{1}{c_0}\frac{\partial \mathbf{H}}{\partial t}, \tag{1,2} \]

\[ \operatorname{div}\mathbf{H}=0, \tag{1,3} \]

\[ \operatorname{div}\mathbf{E}=\frac{4\pi}{\varepsilon_0}\rho_e, \tag{1,4} \]

in which \(\rho_e\) is the electric charge density, \(\mathbf{j}\) the current density, \(c_0\) the electrodynamic constant (the speed of light in vacuum). In equations (1.1)—(1.4) no distinction is made between the magnetic-field strength \(\mathbf{H}\) and the magnetic induction vector \(\mathbf{B}=\mu\mathbf{H}\), since in all known conducting liquids and gases the deviation of the magnetic permeability \(\mu\) from unity is negligibly small. The dielectric permittivity of the medium \(\varepsilon_0\) is assumed to be constant.

The current density \(\mathbf{j}\) is composed of the convection current \(\rho_e\mathbf{v}\) and the conduction current, which also includes the induction current arising when an electrically conducting medium moves with velocity \(\mathbf{v}\) in the magnetic field \(\mathbf{H}\):

\[ \mathbf{j}=\rho_e\mathbf{v}+\sigma\left(\mathbf{E}+\frac{1}{c_0}[\mathbf{v}\mathbf{H}]\right). \tag{1.5} \]

In this expression for the current density it is assumed that the medium has a homogeneous and isotropic conductivity \(\sigma\). The assumption of isotropy of \(\sigma\) substantially restricts the applicability of magnetohydrodynamics to a rarefied gas situated in a strong magnetic field. Indeed, the conductivity of an ionized gas in a magnetic field is isotropic only in the case where the radius of the Larmor orbit of the electrons is considerably greater than their mean free path \(l\), i.e. if \(muc_0/eH \gg l\), where \(m\), \(e\), and \(u\) are the mass, charge, and thermal velocity of the electrons. Using the expressions for the Larmor frequency \(\omega_H=eH/mc_0\) and the mean free time \(\tau=l/u\), we find that the conductivity is isotropic under the condition

\[ \omega_H\tau \ll 1. \tag{1.6} \]

If condition (1.6) is not satisfied, then the conductivity transverse to the magnetic field is, in order of magnitude, \(1+\omega_H^2\tau^2\) times smaller than the longitudinal conductivity \(^{121}\), and the simple expression (1.5) can no longer be used. For this case the anisotropic Ohm’s law replacing expression (1.5) was formulated in work \(^{68}\).

Under the assumption that the conductivity of the medium is large and that electromagnetic processes that are not too rapid are considered, in equations (1.1) and (1.5) one may neglect the displacement current and the convection current in comparison with the conduction current. This assumption, usual for conductors of the metallic type, is fundamental for magnetohydrodynamics \(^{57}\). It is certainly valid under the condition

\[ \frac{\varepsilon_0}{4\pi}\frac{\omega}{\sigma}\ll 1, \tag{1.7} \]

where \(\omega\) is the frequency of the electromagnetic wave or, in the general case, the characteristic inverse time of the process. The conductivity of the media usually considered in magnetohydrodynamics is close to the conductivity of metallic conductors, and therefore relation (1.7) is satisfied up to frequencies close to optical ones. Omitting, by virtue of condition (1.7), the displacement current and the convection current in equations (1.1)—(1.5), we obtain

\[ \mathbf{j}=\frac{c_0}{4\pi}\operatorname{rot}\mathbf{H}, \tag{1.8} \]

\[ \mathbf{E}=-\frac{1}{c_0}[\mathbf{v}\mathbf{H}]+\frac{c_0}{4\pi\sigma}\operatorname{rot}\mathbf{H}, \tag{1.9} \]

\[ \rho_e=-\frac{\varepsilon_0}{4\pi c_0}\operatorname{div}[\mathbf{v}\mathbf{H}], \tag{1.10} \]

\[ \operatorname{div}\mathbf{H}=0, \tag{1.11} \]

\[ \frac{\partial\mathbf{H}}{\partial t}=\operatorname{rot}[\mathbf{v}\mathbf{H}]+\frac{c_0^2}{4\pi\sigma}\nabla^2\mathbf{H}. \tag{1.12} \]

For given \(\mathbf v\) and \(\mathbf H\), the current density \(\mathbf j\), the electric-field strength \(\mathbf E\), and the charge density \(\rho_e\) are completely determined by equations (1.8)—(1.10). Thus the problem reduces to determining the interaction of the magnetic field \(\mathbf H\) and the hydrodynamic velocity field \(\mathbf v\). It is precisely for this reason that the range of questions considered has received the name magnetic hydrodynamics*).

The motion of the medium obeys the usual hydrodynamic equations, which include the volume electromagnetic force \(\mathbf f_e\):

\[ \rho\left[\frac{\partial \mathbf v}{\partial t}+(\mathbf v\nabla)\mathbf v\right] = -\nabla p+\mathbf f_e+\eta\nabla^2\mathbf v+ \left(\zeta+\frac{\eta}{3}\right)\nabla\operatorname{div}\mathbf v, \tag{1.13} \]

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\rho\mathbf v=0. \tag{1.14} \]

Here \(\mathbf v\) and \(\rho\) are the velocity and density of the medium, \(p\) is the pressure, \(\eta\) the viscosity, and \(\zeta\) the second coefficient of viscosity. Equations (1.13), (1.14) may be applied to a rarefied gas only under the condition that the mean free path of the particles \(l\) is small in comparison with the characteristic size of the problem \(L\):

\[ \frac{l}{L}\ll 1. \tag{1.15} \]

Conditions (1.6), (1.7), and (1.15) characterize the domain of applicability of magnetic hydrodynamics.

The force \(\mathbf f_e\) expresses the action of the electromagnetic field on the charge and current bound to the medium and, obviously, is equal to

\[ \mathbf f_e=\rho_e\mathbf E+\frac{1}{c_0}[\mathbf j\mathbf H]. \tag{1.16} \]

From equations (1.8)—(1.10), under condition (1.7), it follows that the first term on the right-hand side of expression (1.16) is of order \(v^2/c_0^2\) in comparison with the second term. Since the macroscopic velocities of the medium are always considerably smaller than the speed of light, in all applications of magnetic hydrodynamics the condition

\[ \frac{v}{c_0}\ll 1, \tag{1.17} \]

is fulfilled, which makes it possible to confine oneself to the nonrelativistic approximation and to omit terms of order \(v^2/c_0^2\).

A relativistic generalization of magnetic hydrodynamics has been undertaken in works \(^{79,136,137}\). As follows from the results of work \(^{79}\), allowance for relativistic effects becomes essential in the case when the density of magnetic energy is comparable with the density of the total energy of the medium, including the rest energy, or, more strictly, under the condition**)

\[ \frac{H^2}{4\pi} > \rho c_0^2. \tag{1.18} \]

For interstellar gas \((\rho\sim 10^{-24}\ \text{g}/\text{cm}^3)\) this corresponds to a magnetic-field strength of the order of \(0.15\) oersted. Since the magnetic fields presently known are considerably smaller than those required by condition (1.18), below we shall confine ourselves to nonrelativistic magnetic hydrodynamics***).

*) Less successful is the shorter term “hydromagnetism,” sometimes used as an equivalent.

**) Under this condition the velocity of magnetohydrodynamic waves (see below) is comparable with the speed of light.

***) In particular, in interstellar space the magnetic field hardly exceeds \(10^{-5}\) oersted. In the other known cases the difference between the fields required by condition (1.18) and the observed ones is still greater.

Omitting, in view of what was said above, the first term in the expression for the electromagnetic force (1.16) and using equation (1.8), we obtain

\[ \mathbf f_e=\frac{1}{4\pi}\,[\operatorname{rot}\mathbf H\cdot \mathbf H]. \tag{1.19} \]

Equations (1.11)—(1.14), together with expression (1.19), determine the behavior of a well-conducting liquid or gaseous medium and of the magnetic field associated with it. Thus, the system of equations of magnetic hydrodynamics has the form:

\[ \frac{\partial \mathbf H}{\partial t} =\operatorname{rot}[\mathbf v\mathbf H]+\nu_m \nabla^2\mathbf H, \tag{1.20} \]

\[ \operatorname{div}\mathbf H=0, \tag{1.21} \]

\[ \frac{\partial \mathbf v}{\partial t}+(\mathbf v\nabla)\mathbf v =-\frac{1}{\rho}\nabla p-\frac{1}{4\pi\rho}[\mathbf H\operatorname{rot}\mathbf H] +\frac{\eta}{\rho}\nabla^2\mathbf v +\frac{1}{\rho}\left(\zeta+\frac{\eta}{3}\right)\nabla\operatorname{div}\mathbf v, \tag{1.22} \]

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\rho\mathbf v=0, \tag{1.23} \]

where

\[ \nu_m=\frac{c_0^2}{4\pi\sigma}. \tag{1.24} \]

The quantity \(\nu_m\) plays in equation (1.20) the same role as the kinematic viscosity \(\nu=\eta/\rho\) in the equation of motion of the medium (1.22), and therefore is often called the magnetic viscosity. Together with the equation of state of the medium, taken, for example, in the form

\[ p=p(\rho,T), \tag{1.25} \]

where \(T\) is the temperature, this system contains two vector and two scalar equations for the quantities \(\mathbf v,\mathbf H,p,\rho\), and \(T\), and must be supplemented by one more equation. The energy equation serves as such an equation\(^{141}\). Since the total energy per unit volume is equal to *)

\[ \frac{\rho v^2}{2}+\rho\varepsilon+\frac{H^2}{8\pi}, \]

where \(\varepsilon\) is the internal energy per unit mass of the medium, the energy equation must have the form

\[ \frac{\partial}{\partial t} \left(\frac{\rho v^2}{2}+\rho\varepsilon+\frac{H^2}{8\pi}\right) =-\operatorname{div}\mathbf g. \tag{1.26} \]

The energy-flux density \(\mathbf g\) is composed of the density of the hydrodynamic energy flux \(\rho\mathbf v\left(\frac{v^2}{2}+w\right)\), where \(w\) is the heat function per unit mass of the medium, the density of the electromagnetic energy flux, expressed by the Umov—Poynting vector \(\frac{c}{4\pi}[\mathbf E\mathbf H]\), which, by virtue of equation (1.9), is equal to \(\frac{1}{4\pi}[\mathbf H[\mathbf v\mathbf H]]-\frac{\nu_m}{4\pi}[\mathbf H\operatorname{rot}\mathbf H]\), and the energy-flux density \(-(\mathbf v\sigma')\), due to processes of internal friction, where

\[ \sigma'_{ik} =\eta\left( \frac{\partial v_i}{\partial x_k} +\frac{\partial v_k}{\partial x_i} -\frac{2}{3}\delta_{ik}\frac{\partial v_l}{\partial x_l} \right) +\zeta\delta_{ik}\frac{\partial v_l}{\partial x_l}. \tag{1.27} \]

*) In view of conditions (1.7), (1.17), the energy of the electric field is negligibly small in comparison with the magnetic energy.

is the “viscous” stress tensor*), and, finally, of the heat-flux density \(-\chi\nabla T\), where \(\chi\) is the thermal-conductivity coefficient of the medium. Thus, the energy-flux density in magnetohydrodynamics is equal to

\[ \mathbf{g} = \rho \mathbf{v}\left(\frac{v^2}{2}+w\right) + \frac{1}{4\pi}[\mathbf{H}[\mathbf{v}\mathbf{H}]] - \frac{\gamma_m}{4\pi}[\mathbf{H}\operatorname{rot}\mathbf{H}] - (\mathbf{v}\boldsymbol{\sigma}) - \chi\nabla T . \tag{1,28} \]

With the aid of equations (1,20)—(1,23) and the thermodynamic identity

\[ dw = T\,ds + \frac{1}{\rho}\,dp \]

equation (1,26) can be transformed into the equation of heat transfer

\[ \rho T\left(\frac{\partial s}{\partial t}+\mathbf{v}\nabla s\right) = \sigma'_{ik}\frac{\partial v_i}{\partial x_k} + \frac{\gamma_m}{4\pi}(\operatorname{rot}\mathbf{H})^2 + \operatorname{div}(\chi\nabla T), \tag{1,29} \]

where \(s\) is the entropy per unit mass. This equation shows that the change in the amount of heat in a moving volume element \((dQ=\rho T\,ds)\) is determined by viscosity, Joule losses, and thermal conduction.

From equations (1,20)—(1,23) it is easy to obtain the law of conservation of momentum for the system under consideration in the form

\[ \frac{\partial \rho v_i}{\partial t} = - \frac{\partial \pi_{ik}}{\partial x_k}, \tag{1,30} \]

where the momentum-flux-density tensor \(\pi_{ik}\) is equal to

\[ \pi_{ik} = p\delta_{ik} + \rho v_i v_k + \frac{1}{4\pi} \left( \frac{H^2}{2}\delta_{ik} - H_iH_k \right) - \sigma'_{ik}. \tag{1,31} \]

The momentum of the electromagnetic field is negligibly small and does not enter into equation (1,30), since, by virtue of conditions (1,7), (1,17), sufficiently slow processes are considered and the displacement current is omitted.

With the aid of a known formula of vector analysis**)

\[ \frac{d}{dt}\iint_{(S)} \mathbf{H}\,d\mathbf{S} = \iint_{(S)} \left\{ \frac{\partial \mathbf{H}}{\partial t} + \mathbf{v}\operatorname{div}\mathbf{H} - \operatorname{rot}[\mathbf{v}\mathbf{H}] \right\} \,d\mathbf{S} \tag{1,32} \]

equation (1,20), taking into account (1,8) and (1,21), can be transformed into the integral form\({}^{54}\):

\[ \frac{d}{dt}\iint_{(S)} \mathbf{H}\,d\mathbf{S} = -\gamma_m \int_{(L)} \operatorname{rot}\mathbf{H}\,d\mathbf{l} = -\frac{c_n}{\sigma}\int_{(L)} \mathbf{j}\,d\mathbf{l}, \tag{1,33} \]

where the surface integral is taken over the “material,” i.e. moving together with the particles of the medium, surface \(S\), bounded by the contour \(L\). From equation (1,33) it follows that the change of magnetic flux through a surface attached to the medium is determined by the electrical resistance of the medium.

The relative importance of the processes of dissipation of the magnetic field due to the finite conductivity of the medium can be estimated as follows. If \(L\) and \(V\) are, respectively, the linear dimension and velocity characteristic for the problem under consideration, then the first term on the right-hand side of equation (1,20), in comparison with

* See, for example, L. D. Landau and E. M. Lifshitz, Mechanics of Continuous Media, Gostekhizdat, 1953, p. 66.

** See, for example, V. I. Smirnov, A Course of Higher Mathematics, vol. 2, p. 344, 1951.

which therefore has the order of magnitude

\[ R_m=\frac{vL}{\nu_m}. \tag{1.34} \]

By analogy with the hydrodynamic Reynolds number \(R=\dfrac{vL}{\nu}\), the dimensionless combination (1.34) is commonly called the magnetic Reynolds number. For \(R_m \gg 1\), the electrical resistance of the medium and the associated Joule losses and dissipation of the magnetic field may be neglected in exactly the same way as viscosity may be neglected in hydrodynamics at large values of \(R\). In laboratory experiments with mercury and liquid sodium \(R_m\sim 10^{-2}-10^0\). In this case the resistance of the medium plays an essential role. Conversely, in astrophysical applications of magnetohydrodynamics \(R_m\sim 10^6\) and higher, owing to the good conductivity of the ionized gas and the enormous dimensions of the objects under consideration. Since, moreover, the ordinary Reynolds numbers are also large, in many problems it is sufficient to restrict oneself to the consideration of an ideal fluid with infinite conductivity. It is then understood that heat-exchange processes are also inessential, i.e. the motion of the medium is adiabatic. Let us note, however, that in a rarefied ionized gas radiative thermal conductivity may play an essential role. In the latter case the motion of the medium should be regarded as isothermal rather than adiabatic\({}^{83}\).

For an ideal medium (\(\eta,\ \zeta,\ \chi\) are equal to zero, the conductivity is infinite), equations (1.20)—(1.23) and (1.30) reduce to the following:

\[ \frac{\partial \mathbf H}{\partial t}=\operatorname{rot}[\mathbf v\mathbf H], \tag{1.35} \]

\[ \operatorname{div}\mathbf H=0, \tag{1.36} \]

\[ \frac{\partial \mathbf v}{\partial t}+(\mathbf v\nabla)\mathbf v =-\frac{1}{\rho}\nabla p-\frac{1}{4\pi\rho}[\mathbf H\operatorname{rot}\mathbf H], \tag{1.37} \]

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\rho\mathbf v=0, \tag{1.38} \]

\[ \frac{\partial s}{\partial t}+(\mathbf v\nabla)s=0. \tag{1.39} \]

Equation (1.39) expresses the conservation of entropy in adiabatic motion of an ideal medium. Together with the equation of state, equations (1.35)—(1.39) form the complete system of magnetohydrodynamic equations for an ideal medium. Equations (1.37)—(1.39) may be written in the form of conservation laws for energy, momentum, and mass:

\[ \frac{\partial}{\partial t}\left(\frac{\rho v^2}{2}+\rho\varepsilon+\frac{H^2}{8\pi}\right) =-\operatorname{div}\mathbf g, \tag{1.40} \]

\[ \frac{\partial \rho v_i}{\partial t} =-\frac{\partial \pi_{ik}}{\partial x_k}, \tag{1.41} \]

\[ \frac{\partial \rho}{\partial t} =-\operatorname{div}\rho\mathbf v, \tag{1.42} \]

where the energy flux density and the tensor of the momentum-flux density are respectively equal to

\[ \mathbf g=\rho\mathbf v\left(\frac{v^2}{2}+w\right) +\frac{1}{4\pi}\left(H^2\mathbf v-(\mathbf H\mathbf v)\mathbf H\right), \tag{1.43} \]

\[ \pi_{ik}=p\delta_{ik}+\rho v_i v_k +\frac{1}{4\pi}\left(\frac{H^2}{2}-H_iH_k\right). \tag{1.44} \]

Equation (1.35) also expresses a certain conservation law that is very characteristic of magnetic hydrodynamics, namely the conservation of the magnetic flux through any surface moving together with the medium[^144]. Indeed, from equations (1.35), (1.36), and (1.32), or directly from equation (1.33) for \(\sigma=\infty\), we have

\[ \frac{d}{dt}\iint \mathbf H\,d\mathbf S=0, \tag{1.45} \]

where the integral is taken over an arbitrary material surface. The conservation of magnetic flux through an arbitrary material surface makes it possible in magnetic hydrodynamics to make extensive use of the visual representation of a magnetic field as a set of lines of force fastened to the medium, or, as they say, “attached” to the medium, “frozen” into it. In fact, if at the initial instant of time there is no flux of the vector \(\mathbf H\) through some material surface, i.e., the lines of force lie on this surface, then, in view of condition (1.45), the lines of force will remain on this surface in the future as well. Since the intersection of two material surfaces possessing this property determines a line of force, the magnetic line of force itself is a material line in the same sense in which we speak of a material surface: it is always connected with definite particles of the medium and moves in the same way as these particles. (Of course, transverse displacements are meant, since any displacement of a magnetic line of force along itself has no meaning.) Owing to this property, any change in the magnetic field may be regarded as its motion, i.e., as the displacement of the lines of force of this field. Everything said is valid only for an infinite conductivity of the medium. An attempt to extend the representation of the motion of the magnetic field to the case of finite conductivity was undertaken in works[^139][^52].

In the case of an incompressible fluid, the equations of magnetic hydrodynamics may be written in a simple symmetric form for the variables[^55]

\[ \mathbf u=\mathbf v+\frac{\mathbf H}{\sqrt{4\pi\rho}}, \qquad \mathbf w=\mathbf v-\frac{\mathbf H}{\sqrt{4\pi\rho}} . \tag{1.46} \]

In these variables*) the system of equations (1.20)—(1.23) takes the form

\[ \left. \begin{aligned} \frac{\partial \mathbf u}{\partial t}+(\mathbf w\nabla)\mathbf u &=-\nabla\Phi+\nabla^2(\alpha\mathbf u+\beta\mathbf w),\\ \frac{\partial \mathbf w}{\partial t}+(\mathbf u\nabla)\mathbf w &=-\nabla\Phi+\nabla^2(\alpha\mathbf w+\beta\mathbf u),\\ \operatorname{div}\mathbf u&=0,\qquad \operatorname{div}\mathbf w=0, \end{aligned} \right\} \tag{1.47} \]

where

\[ \Phi=\frac{p}{\rho}+\frac{(\mathbf u-\mathbf w)^2}{8},\qquad \alpha=\frac{\nu+\nu_m}{2},\qquad \beta=\frac{\nu-\nu_m}{2}. \tag{1.48} \]

The density of the total energy (kinetic plus magnetic), expressed through the variables (1.46), is equal to

\[ \frac{\rho v^2}{2}+\frac{H^2}{8\pi} = \frac{\rho}{4}(u^2+w^2), \tag{1.49} \]

and the difference between the kinetic and magnetic energies is equal to

\[ \frac{\rho v^2}{2}-\frac{H^2}{8\pi} = \frac{\rho}{2}\mathbf u\mathbf w. \tag{1.50} \]

*) Since \(\mathbf v\) is a vector and \(\mathbf H\) a pseudovector, the quantities \(\mathbf u\) and \(\mathbf w\) have no definite transformation properties.

In conclusion of this section we point out that, in magnetic hydrodynamics, Thomson’s classical theorem on the conservation of the circulation of velocity in an ideal fluid[^84] is not satisfied. In the presence of a magnetic field, the circulation of velocity along a material contour is conserved only in the case when the electromagnetic force per unit mass (1.19) has a potential,

\[ \operatorname{rot}\left\{\frac{1}{\rho}[\operatorname{rot}\mathbf H\cdot \mathbf H]\right\}=0, \tag{1.51} \]

which, generally speaking, is not the case.

2. PROPAGATION OF SMALL DISTURBANCES

As a result of the interaction of electromagnetic and hydrodynamic phenomena, small disturbances in a conducting medium in the presence of a magnetic field propagate in the form of waves whose properties differ from those of ordinary sound or electromagnetic waves. First of all, a conducting medium in a magnetic field acquires a characteristic anisotropy: the velocity of wave propagation depends on the direction of propagation with respect to the magnetic field. Moreover, in contrast to sound and electromagnetic waves, waves in magnetic hydrodynamics are, in general, neither longitudinal nor transverse. Waves of small amplitude in a compressible conducting medium in the presence of a magnetic field were first considered in the works[^77],[^80],[^76]. Apart from their independent significance, the study of the behavior of small disturbances has a direct bearing on the study of finite-amplitude waves and, in particular, shock waves in magnetic hydrodynamics.

The possible types of small-amplitude waves in magnetic hydrodynamics are easily established if one restricts oneself to the case of an ideal medium. For generality, let us consider a stationary homogeneous flow of a fluid in a constant magnetic field \(\mathbf H\). It is easy to see that equations (1.35)—(1.39) are satisfied for arbitrary constant \(\mathbf v\) and \(\mathbf H\). Let the initial stationary state be subjected to a small disturbance, as a result of which the velocity, magnetic-field intensity, density, pressure, and entropy undergo small deviations

\[ \mathbf v',\quad \mathbf H',\quad \rho',\quad p' \quad \text{and} \quad s' \tag{2.1} \]

from their stationary values. For what follows it is convenient to introduce the notation

\[ \mathbf u=\frac{\mathbf H}{\sqrt{4\pi\rho}},\qquad \mathbf u'=\frac{\mathbf H'}{\sqrt{4\pi\rho}}, \tag{2.2} \]

where \(\rho\) is the unperturbed density of the medium, constant throughout space. From equations (1.35)—(1.39), neglecting products of the small quantities (2.1) and using the notation (2.2), we obtain the following system of linear equations for the quantities characterizing the small disturbance:

\[ \begin{aligned} \frac{\partial \mathbf u'}{\partial t}+(\mathbf v\nabla)\mathbf u' &=(\mathbf u\nabla)\mathbf v'-\mathbf u\,\operatorname{div}\mathbf v',\\ \operatorname{div}\mathbf u'&=0,\\ \frac{\partial \mathbf v'}{\partial t}+(\mathbf v\nabla)\mathbf v' &=-\frac{1}{\rho}\nabla(p'+\rho\,\mathbf u\mathbf u')+(\mathbf u\nabla)\mathbf u',\\ \frac{\partial \rho'}{\partial t}+(\mathbf v\nabla)\rho' &=-\rho\,\operatorname{div}\mathbf v',\\ \frac{\partial s'}{\partial t}+(\mathbf v\nabla)s'&=0. \end{aligned} \tag{2.3} \]

With the aid of the equation of state, a small pressure perturbation can be expressed in terms of perturbations of density and entropy

\[ p' = c^2 \rho' + b s', \tag{2.4} \]

where \(c=\sqrt{\left(\dfrac{\partial p}{\partial \rho}\right)_s}\) is the speed of sound in the medium in the absence of a magnetic field, and \(b=\left(\dfrac{\partial p}{\partial s}\right)_\rho\). By virtue of the linearity of equations (2.3) and (2.4), an arbitrary perturbation can be represented as a superposition of plane waves with dependence on coordinates and time of the form

\[ e^{i(\mathbf{kr}-\omega t)}, \tag{2.5} \]

where \(\omega\) is the wave frequency and \(\mathbf{k}\) is the wave vector. For plane waves (2.5), equations (2.3) and (2.4) reduce to the following system of algebraic equations:

\[ \begin{gathered} (\omega-\mathbf{k}\mathbf{v})\,\mathbf{u}' + (\mathbf{k}\mathbf{u})\,\mathbf{v}' - \mathbf{u}\,(\mathbf{k}\mathbf{v}') = 0,\\ \mathbf{k}\mathbf{u}' = 0,\\ (\omega-\mathbf{k}\mathbf{v})\,\mathbf{v}' + (\mathbf{k}\mathbf{u})\,\mathbf{u}' - \frac{1}{\rho}\,(p' + \rho\,\mathbf{u}\mathbf{u}')\,\mathbf{k} = 0,\\ (\omega-\mathbf{k}\mathbf{v})\,\rho' - \rho\,(\mathbf{k}\mathbf{v}') = 0,\\ (\omega-\mathbf{k}\mathbf{v})\,s' = 0,\\ p' - c^2\rho' - b s' = 0. \end{gathered} \tag{2.6} \]

Calculating the determinant of the system (2.6) (for this purpose it is convenient to direct one of the coordinate axes along the wave vector \(\mathbf{k}\)) and equating it to zero, we obtain the following condition for the existence of nontrivial solutions:

\[ \omega_0^2\left[\omega_0^2-(\mathbf{k}\mathbf{u})^2\right] \left[\omega_0^4-k^2(c^2+u^2)\omega_0^2+k^2c^2(\mathbf{k}\mathbf{u})^2\right]=0. \tag{2.7} \]

Here

\[ \omega_0=\omega-\mathbf{k}\mathbf{v} \tag{2.8} \]

denotes the frequency in the coordinate system relative to which the fluid is at rest. Therefore \(\dfrac{\omega_0}{k}=V\) is the phase velocity of the wave in the stationary fluid. Equation (2.7) determines four values of \(\omega_0\) distinct in absolute magnitude and, consequently, four distinct waves, each of which has its own velocity of propagation relative to the medium \(V=\dfrac{\omega_0}{k}\). Let us consider in more detail the properties of these waves.

a) Entropy wave

The solution of equation (2.7)

\[ \omega_0=\omega-\mathbf{k}\mathbf{v}=0 \tag{2.9} \]

corresponds to a perturbation stationary relative to the medium. If the medium moves, then a perturbation of this type is transported together with it. Using condition (2.9) in equations (2.6), it is easy to see that in such a wave only the density and entropy are perturbed, related by the condition

\[ \rho'=-\frac{b}{c^2}s'. \tag{2.10} \]

The remaining quantities are unchanged:

\[ \mathbf{v}'=0,\qquad \mathbf{u}'=0,\qquad p'=0. \tag{2.11} \]

Such perturbations coincide with the entropic waves known in ordinary hydrodynamics. They can be called waves only conditionally, since the velocity of propagation of these perturbations relative to the medium is equal to zero. Nevertheless, in a number of cases, for example in studying the behavior of shock waves under small perturbations, the entropic wave must be taken into account alongside the true waves considered below.

b) Magnetohydrodynamic wave

Equation (2.7) has the solution

\[ \omega_0=\pm(\mathbf{k}\mathbf{u}), \tag{2.12} \]

which corresponds to waves propagating with velocity

\[ V_A=\pm \frac{H}{\sqrt{4\pi\rho}}\cos\vartheta, \tag{2.13} \]

where \(\vartheta\) is the angle between the direction of propagation of the wave and the magnetic-field strength. Substituting (2.12) into equations (2.6), it is not difficult to see that in such waves the thermodynamic characteristics of the medium remain unchanged,

\[ \rho'=0,\qquad p'=0,\qquad s'=0, \tag{2.14} \]

while the perturbations of the velocity and of the magnetic-field strength obey the conditions

\[ \mathbf{v}'=\mp \mathbf{u}',\qquad \mathbf{k}\mathbf{u}'=0,\qquad \mathbf{u}\mathbf{u}'=0. \tag{2.15} \]

By virtue of conditions (2.15), waves of the type under consideration are purely transverse, with the oscillations occurring in a direction perpendicular to the unperturbed magnetic field \(\mathbf{H}\). These waves have no analogue in ordinary hydrodynamics. They are specific to magnetic hydrodynamics and have been given the name magnetohydrodynamic waves. The transversality of these waves means that, in a magnetic field, a conducting medium acquires a peculiar elasticity with respect to shear of its neighboring layers. This is one of the essential features of magnetic hydrodynamics.

Magnetohydrodynamic waves are not accompanied by a change in density and therefore are possible both in a compressible and in an incompressible medium. For an incompressible medium the existence of magnetohydrodynamic waves was theoretically predicted by Alfvén \(^{1,2,5}\). Subsequently, magnetohydrodynamic waves were detected in experiments with mercury \(^{100,101}\), liquid sodium \(^{94}\), and a gas discharge \(^{16,17}\). The discovery of a new kind of wave motion was one of the most important results of magnetic hydrodynamics and found numerous applications, above all to a number of problems in astrophysics. In the literature, magnetohydrodynamic waves are often also called Alfvén waves.

From the point of view of the microscopic theory of an ionized gas, magnetohydrodynamic waves are a limiting case of ordinary electromagnetic waves in an ionized gas, corresponding to frequencies much smaller than the ion gyrofrequency:

\[ \omega \ll \Omega=\frac{eH}{mc}, \tag{2.16} \]

where \(e\) and \(m\) are the charge and mass of the heavy ions. In this case, taking into account the motion of the ions in the field of the electromagnetic wave in the presence of an external magnetic field \(H\) leads to expression (2.13) for the propagation velocity of the electromagnetic wave \(^{8,9,70}\)*).

* For more details on the propagation of electromagnetic waves in an ionized gas, see \(^{69}\).

c) Magnetosonic waves

In addition to the solutions considered above, the characteristic equation (2.7) is satisfied under the condition

\[ \omega_0^4-k^2(u^2+c^2)\omega_0^2+k^2c^2(\mathbf{k}\mathbf{u})^2=0. \tag{2.17} \]

The solutions of the biquadratic equation (2.17) are two values of \(\omega_0\) different in absolute value. They correspond to two different waves, whose velocities \(V_+\) and \(V_-\) are determined, by virtue of (2.17), by the expression

\[ V_\pm^2=\frac{\omega_0^2}{k^2} =\frac12\left\{c^2+u^2\pm \sqrt{(c^2+u^2)^2-4c^2u^2\cos^2\vartheta}\right\}. \tag{2.18} \]

Here, as before, \(\vartheta\) is the angle between the direction of propagation of the wave and the magnetic-field intensity \(\mathbf{H}\).

The velocities \(V_+\) and \(V_-\), which differ by the sign of the radical in expression (2.18), satisfy the following conditions:

\[ \max(u^2,c^2)\leq V_+^2\leq c^2+u^2, \tag{2.19} \]

\[ 0\leq V_-^2\leq \min(u^2\cos^2\vartheta,c^2). \tag{2.20} \]

Here \(\max\) and \(\min\) denote, respectively, the largest and the smallest of the quantities in parentheses. Since \(c\) is the speed of sound in the absence of a magnetic field, and \(u\cos\vartheta=\dfrac{H}{\sqrt{4\pi\rho}}\cos\vartheta\) is the velocity of the magnetohydrodynamic wave in the given direction, the solution \(V_+\) corresponds to an accelerated wave, and the solution \(V_-\) to a slowed wave in comparison with the sound or magnetohydrodynamic waves.

Under condition (2.17), it follows from equations (2.6) that the entropy of the medium is unchanged \((s'=0)\), while all the remaining quantities may be expressed through the density perturbation:

\[ \mathbf{v}'=-\frac{\omega_0}{\rho k^2}\, \frac{k^2(\mathbf{k}\mathbf{u})\mathbf{u}-\omega_0^2\mathbf{k}} {\omega_0^2-(\mathbf{k}\mathbf{u})^2}\,\rho', \tag{2.21} \]

\[ \mathbf{u}'=\frac{\omega_0^2}{\rho k^2}\, \frac{k^2\mathbf{u}-(\mathbf{k}\mathbf{u})\mathbf{k}} {\omega_0^2-(\mathbf{k}\mathbf{u})^2}\,\rho', \tag{2.22} \]

\[ p'=c^2\rho'. \tag{2.23} \]

Expressions (2.21) and (2.22) show that the perturbations of the velocity and of the magnetic-field intensity, \(\mathbf{v}'\) and \(\mathbf{u}'\), unlike in a magnetohydrodynamic wave, lie in the plane \((\mathbf{k},\mathbf{H})\) and have components both in the direction of propagation of the wave and in the perpendicular direction. This means that the accelerated and slowed magnetosonic waves are neither longitudinal nor transverse*).

Let us note that, by virtue of expressions (2.2), (2.22), and (2.23), the perturbation of the magnetic pressure \(p_m=\dfrac{H^2}{8\pi}\) is equal to

\[ p_m'=\rho\mathbf{u}\mathbf{u}'=\left(\frac{V^2}{c^2}-1\right)p'. \tag{2.24} \]

It follows from this that, for the accelerated wave \((V_+^2>c^2)\), \(p_m'\) and \(p'\) have the same signs and, consequently, the Maxwell stresses of the magnetic field act in the same direction as the pressure perturbation. For the slowed wave \((V_-^2<c^2)\), \(p_m'\) and \(p'\) have opposite signs and partially compensate one another.

* In an ionized gas, magnetosonic waves are the limiting case of electromagnetic waves with frequencies satisfying (2.16)\(^{69}\).

The vector diagram of the velocities determined by expressions (2.13) and (2.18) is shown in Fig. 1 for two cases: \(u=0.9c\) and \(u=1.11c\). The horizontal axis corresponds to the direction of the applied magnetic field. The velocity of propagation of disturbances in an arbitrary direction making an angle \(\vartheta\) with the direction of the magnetic field is determined by the length of the vector drawn at this angle from the origin to its intersection with the corresponding curve.

For \(\vartheta=0\), the accelerated wave becomes an ordinary sound wave if \(c>u\), or a magnetohydrodynamic wave if \(u>c\). Under the same conditions, the retarded wave becomes, respectively, a magnetohydrodynamic or a sound wave. For \(\vartheta=\dfrac{\pi}{2}\), the propagation velocity of the magnetohydrodynamic and retarded magnetosonic waves becomes

Fig. 1.

zero. In this case both waves reduce to a weak tangential discontinuity, in which the perturbations of velocity and field are parallel to the plane of the front.

It follows from equations (2.13) and (2.18) that the velocities \(V_+\), \(V_A\), and \(V_-\) do not depend on the wave frequency \(\omega\), i.e. in an ideal medium there is no dispersion. Dispersion appears when wave damping due to the finite conductivity of the medium is taken into account, as was shown for plane magnetohydrodynamic waves in work \(^{125}\)*), and for cylindrical (torsional) magnetohydrodynamic waves in work \(^{101}\), in connection with the analysis of the experiments carried out in that work, which made it possible to realize torsional magnetohydrodynamic waves experimentally in mercury. In work \(^{6}\), the longitudinal magnetosonic wave (the wave \(V_+\) for \(\vartheta=\dfrac{\pi}{2}\), propagating in a direction perpendicular to the magnetic field) is studied in detail, taking into account weak damping due to the finite conductivity of the medium. The dependence of the damping on the wavelength here also leads to dispersion.

The propagation of magnetohydrodynamic waves in a rotating incompressible liquid was considered in works \(^{93,95}\). The presence of the Coriolis force leads to the appearance of two circularly polarized magnetohydrodynamic waves with opposite directions of rotation

*) See also \(^{117}\).

and different phase velocities. The influence of the Coriolis force is characterized by the dimensionless parameter \(\Omega/kV_A\), where \(\Omega\) is the angular velocity of rotation and \(k\) is the wave number. In view of the dependence of the effect on the wavelength, dispersion also occurs in this case.

A more general problem on the propagation of plane and cylindrical (torsional) waves in a medium with finite conductivity, taking into account displacement currents, was considered in papers \(^{10,11}\) in the limiting cases of weak damping and a weak magnetic field.

In paper \(^{78}\) the propagation of disturbances in an incompressible viscous fluid is investigated, the density of which is inhomogeneous in the direction of gravity and of a parallel magnetic field. With the aid of the variational method, developed for a layer of fluid of finite thickness, approximate solutions are obtained in the limiting cases of homogeneous density and infinite conductivity.

The reflection and refraction of plane magnetohydrodynamic waves at a plane interface between two media of different densities is considered in paper \(^{64}\) for magnetohydrodynamic waves polarized perpendicular to the plane of incidence, and in paper \(^{124}\) for arbitrary polarization*).

3. DISCONTINUITY SURFACES AND SHOCK WAVES

As do the ordinary hydrodynamic equations, the equations of magnetic hydrodynamics for an ideal medium \((\eta=\zeta=\chi=0,\ \sigma=\infty)\) admit discontinuous solutions in which the characteristics of the medium and of the field undergo a jump-like change on certain surfaces. In ordinary hydrodynamics there exist two types of such discontinuity surfaces: the tangential discontinuity and the shock wave. In magnetic hydrodynamics the picture becomes considerably more complicated. Shock waves in magnetic hydrodynamics were first considered by Hoffmann and Teller \(^{79}\)**), starting from the relativistic energy–momentum tensor for the medium and the electromagnetic field. As follows from this work, a relativistic treatment is necessary only in the case where the density of magnetic energy is, in order of magnitude, comparable with the density of the total energy of the medium, including rest energy (cf. (1.18)). In all practically important cases the energy of the magnetic field is much less than the total energy of the medium; therefore only nonrelativistic shock waves will be considered below.

The types of discontinuity surfaces possible in magnetic hydrodynamics can be found by considering the boundary equations on the discontinuity surface \(^{1,10,141}\). The latter are easily obtained from equations (1.35), (1.36), and (1.40–1.42), using a coordinate system in which the discontinuity surface is at rest and coincides with the plane \((y,z)\). In this case the complete system of boundary equations on a magnetohydrodynamic discontinuity surface has the form

\[ \left. \begin{array}{lll} \{[\mathbf{v}\mathbf{H}]_y\}=0, & \{[\mathbf{v}\mathbf{H}]_z\}=0, & \{H_x\}=0,\\ \{g_x\}=0, & \{\rho v_x\}=0, & \{\pi_{xi}\}=0 \quad (i=x,y,z). \end{array} \right\} \tag{3,1} \]

Here and in what follows in this section, curly braces denote the difference of the values, on the two sides of the discontinuity surface, of the quantity enclosed in them. The meaning of conditions (3.1) is evident. The first three of them are the usual electrodynamic continuity conditions—

* See also the review article \(^{416}\).

** See the collection \(^{109}\).

of the tangential component of the electric field and the normal component of the magnetic field, since for a medium with infinite conductivity it follows from equation (1.9) that

\[ \mathbf{E}=-\frac{1}{c_0}[\mathbf{v}\mathbf{H}]. \tag{3.2} \]

The last three of equations (3.1) express, respectively, the continuity of the fluxes of energy, mass, and momentum. Using expressions (1.43) and (1.44) for the densities of the fluxes of energy and momentum, and expanding the vector product \([\mathbf{v}\mathbf{H}]\), we obtain the following boundary equations for a discontinuity surface in magnetohydrodynamics:

\[ \{v_xH_y-v_yH_x\}=0,\qquad \{v_xH_z-v_zH_x\}=0, \tag{3.3} \]

\[ \{H_x\}=0, \tag{3.4} \]

\[ \left\{\rho v_x\left(\frac{v^2}{2}+w\right)+ \frac{1}{4\pi}\left(H^2v_x-(\mathbf{v}\mathbf{H})H_x\right)\right\}=0, \tag{3.5} \]

\[ \{\rho v_x\}=0, \tag{3.6} \]

\[ \left\{p+\rho v_x^2+\frac{H^2}{8\pi}\right\}=0, \tag{3.7} \]

\[ \left\{\rho v_xv_y-\frac{1}{4\pi}H_xH_y\right\}=0,\qquad \left\{\rho v_xv_z-\frac{1}{4\pi}H_xH_z\right\}=0. \tag{3.8} \]

For \(H=0\), equations (3.3)—(3.8) reduce to the ordinary hydrodynamic equations on a discontinuity surface, which admit only two mutually exclusive types of discontinuities: either \(\{v_y\}\ne0\), while \(v_x=0\)—a tangential discontinuity, or \(v_x\ne0\), while \(\{v_y\}=0\)—a shock wave. Together with the existence of a minimum speed of propagation of a shock wave (equal to the speed of sound), this means, in particular, that in hydrodynamics small perturbations cannot transform a tangential discontinuity into a shock wave, and conversely. Therefore the introduction of two types of discontinuities in ordinary hydrodynamics has a deep physical basis. The situation is different in magnetic hydrodynamics. Equations (3.3)—(3.8) do not split into separate mutually exclusive groups and, as was shown in paper \(^{141}\), any of the discontinuities allowed by these equations can, generally speaking, pass into any other under a continuous change of the conditions of motion. This means that any classification of magnetohydrodynamic discontinuities, including the one given below, is only conventional. Nevertheless, such a classification is very convenient in investigating the properties of discontinuities in magnetohydrodynamics and is widely used in the literature \(^{79,\,76,\,105*}\). It is based on the external features of the motion near the discontinuity surface.

a) Tangential discontinuity

This type includes discontinuities in which the normal component of the velocity is absent,

\[ v_x=0, \tag{3.9} \]

i.e. the discontinuity surface is at rest relative to the fluid. If, in this case, \(H_x\ne0\), then from equations (3.3)—(3.8) follows the continuity

\[ \text{*) See collection }^{109}. \]

velocity, pressure, and magnetic-field intensity. Such a discontinuity can represent only the boundary separating two different media. We shall assume that

\[ H_x=0. \tag{3.10} \]

In this case the velocity and the magnetic-field intensity are parallel to the discontinuity surface and, by virtue of conditions (3.3), (3.5), (3.8), may undergo arbitrary jumps in magnitude and direction. The pressure and the magnetic-field intensity are related by the condition

\[ \left\{p+\frac{H^2}{8\pi}\right\}=0, \tag{3.11} \]

which signifies the continuity of the total pressure \(P=p+\dfrac{H^2}{8\pi}\), composed of the ordinary pressure \(p\) and the “lateral” stress of the magnetic field \(\dfrac{H^2}{8\pi}\). Conditions (3.9), (3.10), and (3.11) completely determine the tangential discontinuity. Such a discontinuity is possible both in a compressible and in an incompressible medium.

b) Perpendicular shock wave

In discontinuities of this type

\[ v_x\ne 0,\qquad H_x=0. \tag{3.12} \]

Under these conditions, as follows from equations (3.6) and (3.8), the tangential component of the velocity must be continuous

\[ \{v_y\}=0,\qquad \{v_z\}=0. \tag{3.13} \]

Consequently, one may pass to such a coordinate system in which on both sides of the discontinuity the tangential component of the velocity is absent and, in addition, the magnetic field is directed along the \(y\)-axis:

\[ v=v_x,\qquad H=H_y. \tag{3.14} \]

In such a coordinate system the perpendicular shock wave, as follows from equations (3.3)—(3.8), obeys the following conditions:

\[ \left. \begin{array}{c} \left\{\dfrac{H}{\rho}\right\}=0,\qquad \{\rho v\}=0,\\[6pt] \left\{\dfrac{v^2}{2}+w+\dfrac{H^2}{4\pi\rho}\right\}=0,\qquad \left\{p+\rho v^2+\dfrac{H^2}{8\pi}\right\}=0. \end{array} \right\} \tag{3.15} \]

A discontinuity of this type is a longitudinal compression shock wave whose direction of propagation is perpendicular to the direction of the magnetic field. The first of conditions (3.15) expresses the “frozen-in” character of the magnetic field in the medium: the quantity \(H/\rho\) is preserved across the discontinuity. The remaining equations, by means of the substitution

\[ \varepsilon^*=\varepsilon+\frac{H^2}{8\pi\rho},\qquad p^*=p+\frac{H^2}{8\pi} \tag{3.16} \]

reduce to the ordinary equations of a shock wave, in which the energy

and the pressure contain the corresponding magnetic terms. The character of the motion in such a discontinuity is shown in Fig. 2.

For \(H=0\) the perpendicular shock wave degenerates into an ordinary one. For \(H\ne 0\) its propagation velocity depends on the magnetic-field strength. As was shown in \(^{79}\), a perpendicular shock wave is possible only as a compression wave. The magnetic field decreases the compressibility of the medium and accordingly increases the propagation velocity of the discontinuity. The dependence between the parameters of this discontinuity has been studied in detail in \(^{76,107}\).

A perpendicular shock wave of weak intensity coincides with an accelerated magnetosonic wave propagating across the magnetic field \(\left(\vartheta=\frac{\pi}{2}\right.\) in Fig. 1\()\) with velocity determined from equation (2,18):

\[ V_{+}^{2}=c^{2}+u^{2}. \tag{3,17} \]

When studying discontinuities for which

\[ v_x\ne 0,\qquad H_x\ne 0, \tag{3,18} \]

it is convenient to use a coordinate system in which the vectors \(\mathbf v\) and \(\mathbf H\) are parallel \(^{79}\). By virtue of equations (3,3) this can be done at once for both sides of the discontinuity surface if one passes to a coordinate system moving relative to the original one parallel to the discontinuity surface with velocity

\[ \mathbf U=\mathbf v-\frac{v_x}{H_x}\mathbf H, \tag{3,19} \]

where \(\mathbf v\) is the velocity of the medium in the original coordinate system. Let us denote by the indices 1 and 2 the quantities referring to the different sides of the discontinuity surface. Then in the chosen coordinate system

\[ \mathbf v_1=q_1\mathbf H_1,\qquad \mathbf v_2=q_2\mathbf H_2, \tag{3,20} \]

where \(q_1\) and \(q_2\) are certain coefficients of proportionality. In this coordinate system the streamlines of the fluid are parallel to the magnetic lines of force and undergo the same refraction with them at the discontinuity surface, as is shown in Fig. 3. Let us note that for a perpendicular shock wave and, in the general case, for a tangential discontinuity the choice of such a coordinate system is impossible.

Fig. 2.

Fig. 3.

Under the conditions (3,18) and (3,20), the boundary equations (3,3)—(3,8) reduce to the following:

\[ \{H_x\}=0, \tag{3,21} \]

\[ \left\{\frac{v^{2}}{2}+w\right\}=0, \tag{3,22} \]

\[ \{\rho v_x\}=0, \tag{3,23} \]

\[ \left\{p+\rho v_x^{2}+\frac{H^{2}}{8\pi}\right\}=0, \tag{3,24} \]

\[ \left\{\rho v_x v_y-\frac{1}{4\pi}H_xH_y\right\}=0,\qquad \left\{\rho v_x v_z-\frac{1}{4\pi}H_xH_z\right\}=0. \tag{3,25} \]

From these equations and relation (3.20) it follows that

\[ \{pq\}=0, \tag{3.26} \]

\[ \left\{\left(1-\frac{1}{4\pi\rho q^2}\right)v_y\right\}=0,\qquad \left\{\left(1-\frac{1}{4\pi\rho q^2}\right)v_z\right\}=0. \tag{3.27} \]

Equations (3.21)—(3.25) and the equations following from them, (3.26), (3.27), admit two essentially different types of discontinuities, depending on whether the density of the medium is continuous or undergoes a jump at the surface of discontinuity.

c) Magnetohydrodynamic wave

If the density of the medium is continuous,

\[ \{\rho\}=0, \tag{3.28} \]

then, by virtue of equation (3.26), the quantity \(q\) is also continuous. Together with equation (3.27) this means that \(q=\pm \dfrac{1}{\sqrt{4\pi\rho}}\), since at least one of the quantities \(\{v_y\}\) or \(\{v_z\}\) is nonzero. Thus, in a magnetohydrodynamic wave the velocity vector is related to the magnetic-field intensity in the chosen coordinate system by the relations

\[ \mathbf{v}_1=\pm \frac{\mathbf{H}_1}{\sqrt{4\pi\rho}},\qquad \mathbf{v}_2=\pm \frac{\mathbf{H}_2}{\sqrt{4\pi\rho}}, \tag{3.29} \]

and at the surface of discontinuity the following conditions are satisfied:

\[ \{v_x\}=0,\qquad \{H_x\}=0,\qquad \{\varepsilon\}=0,\qquad \{\rho\}=0, \tag{3.30} \]

\[ \left\{p+\frac{H_y^2+H_z^2}{8\pi}\right\}=0. \tag{3.31} \]

Here the expression \(w=\varepsilon+\dfrac{p}{\rho}\) has been used for the heat function. From relations (3.29) it follows that the velocity of propagation of the magnetohydrodynamic wave relative to the medium is equal to

\[ V=\mp \frac{1}{\sqrt{4\pi\rho}}H_x. \tag{3.32} \]

A characteristic feature of such a discontinuity is that the medium, when passing through the surface of discontinuity, may acquire an arbitrary tangential impulse in direction, so that in the general case the motion is not plane.

In connection with conditions (3.30), (3.31) it should be noted that, since on both sides of the discontinuity the density and internal energy of the medium are the same, all the remaining thermodynamic parameters, including the pressure \(p\), must also be the same. This means that, in a medium with a single-valued equation of state, condition (3.31) reduces to the following two:

\[ \{p\}=0,\qquad \{H_y^2+H_z^2\}=0, \tag{3.33} \]

i.e., along with the continuity of all thermodynamic parameters, the normal components and the absolute values of the tangential components of the magnetic field and velocity are also continuous. For given values \(\mathbf{H}_1\) and \(\mathbf{v}_1\), the possible values \(\mathbf{H}_2\) and \(\mathbf{v}_2\) lie on the surface

cone whose generator makes with the normal the same angle as the vector \(\mathbf H_1\), as is shown in Fig. 4.

In an incompressible fluid the conditions (3.33) do not follow from conditions (3.30), (3.31), since the pressure is no longer determined by the density and internal energy. The tangential components may undergo an arbitrary jump, related to the pressure jump by condition (3.31). In what follows we shall see that a discontinuous magnetohydrodynamic wave is a special case of the general solution first found by Alfvén\(^5\).

г) Oblique shock wave

Discontinuities of this type are accompanied by a jump in density

\[ \{\rho\}\ne 0. \tag{3.34} \]

Fig. 4.

It can be verified that the motion in this case must be plane, i.e., the coordinate system can be chosen so that

\[ v_z=0,\qquad H_z=0. \tag{3.35} \]

Indeed, by virtue of conditions (3.20), by rotating the coordinate system about the \(x\)-axis one can always make \(v_{1z}=0\) and \(H_{1z}=0\). Then from equations (3.27) it follows either that \(v_{2z}=0\) and \(H_{2z}=0\), i.e. we directly obtain conditions (3.35), or

\[ q_2^2=\frac{1}{4\pi\rho_2},\qquad q_1^2\ne \frac{1}{4\pi\rho_1}, \tag{3.36} \]

(the simultaneous equality \(q_2^2=\dfrac{1}{4\pi\rho_2}\) and \(q_1^2=\dfrac{1}{4\pi\rho_1}\) contradicts conditions (3.26) and (3.34)). In the latter case it follows from conditions (3.27) that \(v_{1y}=0\) and \(H_{1y}=0\), i.e. on one side of the discontinuity the tangential components of the field and velocity are entirely absent, and the coordinate system remains to be chosen so that \(v_{2z}=0\) and \(H_{2z}=0\).

By virtue of conditions (3.35), the boundary equations (3.21)—(3.25) for an oblique shock wave take the form:

\[ \begin{gathered} \{H_x\}=0,\qquad \{\rho v_x\}=0,\qquad \left\{w+\frac{v^2}{2}\right\}=0,\\[6pt] \left\{p+\rho v_x^2+\frac{H_y^2}{8\pi}\right\}=0,\qquad \left\{\rho v_xv_y-\frac{1}{4\pi}H_xH_y\right\}=0. \end{gathered} \tag{3.37} \]

In discontinuities of this type the compression shock wave interacts in a complicated manner with the magnetic field. The dependence between the parameters determining the state of the medium before and after passage of the oblique shock wave was calculated in works \(^{76,107}\) under the assumption that the equation of state of an ideal gas is valid for the medium. For each value of the magnetic-field strength, generally speaking, three different shock waves of this type may be realized.

In the case \(H_y=0\), but \(H_x\ne0\), the oblique shock wave becomes the so-called parallel shock wave \(^{79}\), propagating along the magnetic field and not interacting with it. In this case

the connection between the hydrodynamic motion and the magnetic field is absent and, as is easy to see from equations (3.37), for \(H_y=0\) the shock wave obeys the usual hydrodynamic equations

\[ \{p+\rho v_x^2\}=0,\qquad \left\{w+\frac{v^2}{2}\right\}=0,\qquad \{\rho v_x\}=0. \tag{3.38} \]

Let us note separately the discontinuity satisfying conditions (3.36). In such a discontinuity

\[ v_{1y}=0,\qquad H_{1y}=0,\qquad v_{2x}=\pm \frac{H_{3x}}{\sqrt{4\pi\rho_2}},\qquad v_{2y}=\pm \frac{H_{3y}}{\sqrt{4\pi\rho_2}} \tag{3.39} \]

and the conditions

\[ \begin{aligned} H_{1x}&=H_{2x},\\ \rho_1 v_{1x}&=\rho_2 v_{2x},\\ w_1+\frac{v_1^2}{2}&=w_2+\frac{v_2^2}{2},\\ p_1+\rho_1 v_{1x}^2&=p_2+\rho_2 v_{2x}^2+\frac{\rho_2 v_{2y}^2}{2}. \end{aligned} \tag{3.40} \]

On one side of such a discontinuity the tangential components \(\mathbf v\) and \(\mathbf H\) are absent, and the motion proceeds as in a parallel shock wave; on the other side the motion proceeds as in a magnetohydrodynamic wave, with arbitrary tangential components of the field and, in accordance with conditions (3.39), of the velocity being admissible. The character of the motion in such a discontinuity is shown in Fig. 5.

Fig. 5.

In the limiting case of discontinuities of weak intensity, oblique shock waves degenerate into one of the magnetosonic waves considered in Section 2—accelerated or retarded.

In work \({}^{14}\) it was shown that continuous transitions between discontinuities of different types can occur according to the following scheme:

\[ \begin{array}{ccc} \text{tangential} & \longleftrightarrow & \text{magnetohydro-}\\ \text{discontinuity} & & \text{dynamic wave}\\[0.5ex] \uparrow & & \uparrow\\[-0.5ex] \multicolumn{3}{c}{\longleftarrow\ \text{oblique shock}\ \longrightarrow}\\ \multicolumn{3}{c}{\text{wave}}\\[-0.5ex] \multicolumn{3}{c}{\downarrow}\\ \multicolumn{3}{c}{\text{perpendicular}}\\ \multicolumn{3}{c}{\text{shock wave}} \end{array} \]

Such transitions occur through certain discontinuities which may be called transitional and which simultaneously satisfy the boundary equations for two adjacent types of discontinuities, i.e., they may be assigned both to one type and to the other. The presence of such transitional discontinuities also means that a discontinuity of one type can pass into a discontinuity of another type under a continuous change of the parameters.

The existence of transitions between discontinuities of different types is easily verified in the example of discontinuities of weak intensity considered in Section 2. Fig. 1 and equation (2.24) show that an accelerated magnetosonic wave at \(\vartheta=\dfrac{\pi}{2}\) passes into a weak perpendicular shock wave, while at \(\vartheta=0\) it passes into a sound wave or into a magnetohydrodynamic one. A decelerated magnetosonic wave at \(\vartheta=0\) coincides with a magnetohydrodynamic wave or with a sound wave, if the velocity of the latter is smaller, and at \(\vartheta=\dfrac{\pi}{2}\) it is a weak tangential discontinuity whose propagation velocity is zero. In exactly the same way, a magnetohydrodynamic wave at \(\vartheta=\dfrac{\pi}{2}\) is simply a tangential discontinuity, while at \(\vartheta=0\) it coincides with one of the two magnetosonic waves.

The concept of a discontinuity surface is a known idealization of the real flow of a medium. In reality, a discontinuity is a finite region of rapid change of the parameters characterizing the state of the medium. The width of this region is determined by dissipative processes caused by viscosity, thermal conductivity, and, in the presence of a magnetic field, by the finite conductivity of the medium. The finite conductivity of the medium makes impossible a sharp jump of the magnetic-field strength and leads to dissipation of magnetic energy in the form of Joule heat.

Thus, for example, a tangential discontinuity under the influence of viscosity and finite conductivity must become smeared out with time. With the aid of equations (1.19) and (1.21) it is easy to show that, at a distance \(L\) from the place where the tangential discontinuity arises, the change in velocity occurs in a layer of thickness \(\delta_v\)

\[ \delta_v \sim \frac{L}{\sqrt{R}}, \tag{3.41} \]

and the change in magnetic-field strength in a layer of thickness

\[ \delta_m \sim \frac{L}{\sqrt{R_m}}, \tag{3.42} \]

where \(R=\dfrac{VL}{\nu}\) and \(R_m=\dfrac{VL}{\nu_m}\) are the ordinary and magnetic Reynolds numbers. In many astrophysical applications of magnetohydrodynamics the numbers \(R\) and \(R_m\) are very large, which makes it possible to neglect the smearing of the tangential discontinuity.

The structure of shock waves in magnetic hydrodynamics has at present been investigated only for the perpendicular shock wave. In work \({}^{141}\) an expression was obtained for the width of a perpendicular shock wave of weak intensity, which shows that in a magnetic field the width of the discontinuity is no longer necessarily of the order of the mean free path: at small conductivity it may considerably exceed the latter.

A detailed investigation of the structure of a perpendicular shock wave in a medium for which the equation of state of an ideal gas is valid was carried out in work \({}^{110}\), taking into account the dependence of viscosity, thermal conductivity, and conductivity on temperature. If the conductivity of the medium is high, then the width of the discontinuity is several mean free paths. In the case of small conductivity the structure of the discontinuity depends substantially on the magnetic-field strength. At small values of \(H\), ahead of the discontinuity there is a region in which the magnetic field, the velocity

and the temperature of the medium vary smoothly. For values of \(H\) greater than a certain critical value, the sharp jump is altogether absent: all quantities vary smoothly over an extended region. This phenomenon is analogous to the isothermal jump known in ordinary hydrodynamics and is connected with the fact that the characteristic lengths are different for Joule dissipation in the discontinuity and for viscous dissipation (i.e., the mean free path).

In work \(^{133}\) the structure of a perpendicular shock wave in a plasma with infinite conductivity is investigated. As shown in works \(^{110,111}\), in an ionized gas, because of the different roles of ions and electrons (viscosity is due mainly to the former, thermal conductivity to the latter) and the large difference in their masses, the coefficient of thermal conductivity substantially exceeds the coefficient of viscosity. Therefore, even in the absence of a magnetic field there exist conditions for the occurrence of an ordinary isothermal jump.

4. SOME SOLUTIONS OF THE EQUATIONS OF MAGNETOHYDRODYNAMICS

a) Problems of magnetohydrostatics

A static magnetic field in a motionless medium in the absence of external sources is possible only if the conductivity of the medium is infinitely large, i.e., the magnetic field is “frozen” into the medium. Finite conductivity, by virtue of equation (1.20), would lead to dissipation of the magnetic field, accompanied by conversion of its energy into Joule heat. Thus, magnetohydrostatics studies the equilibrium conditions of a conducting medium under the action of pressure forces, the tension of the magnetic field “frozen” into the medium and, when necessary, taking them into account, gravitational forces.

In the static case \(\left(\dfrac{\partial}{\partial t}=0,\ \mathbf{v}=0\right)\) the system of magnetohydrodynamic equations reduces to the equations

\[ \frac{1}{4\pi}[\mathbf{H}\operatorname{rot}\mathbf{H}]=-\nabla p, \tag{4.1} \]

\[ \operatorname{div}\mathbf{H}=0. \tag{4.2} \]

It follows from equation (4.1) that, for hydrostatic equilibrium of a conducting medium in a magnetic field, it is necessary that the Lorentz force be everywhere balanced by the pressure gradient. In this case the lines of force of the magnetic field and the current lines \(\mathbf{j}=\dfrac{c_{0}}{4\pi}\operatorname{rot}\mathbf{H}\) are located on the surfaces \(p=\mathrm{const}\) \(^{102}\). If it is necessary to take gravitational forces into account, the term \(\rho\nabla U\), where \(U\) is the gravitational potential, must be added to the right-hand side of equation (4.1). Since the Lorentz force is perpendicular to the magnetic-field intensity, the change of pressure along the lines of force is entirely determined by the gravitational forces.

In connection with theories of stellar magnetism, magnetic fields that exert no mechanical action on the medium are of interest, i.e., those for which the Lorentz force in the hydrodynamic equation of motion vanishes:

\[ \frac{1}{4\pi}[\operatorname{rot}\mathbf{H}\cdot \mathbf{H}]=0. \tag{4.3} \]

This means that the current is everywhere parallel to the magnetic field:

\[ \operatorname{rot}\mathbf{H}=\alpha(\mathbf{r})\mathbf{H}. \tag{4.4} \]

Such fields have been called force-free fields[^1]. A particular solution of equation (4.4) is the helical field in an infinitely long cylinder indicated in [^102], which in cylindrical coordinates \(\rho,\varphi,z\) has the form

\[ H_z = A J_0(\alpha \rho), \qquad H_\varphi = A J_1(\alpha \rho), \tag{4.5} \]

where \(A\) and \(\alpha\) are constants, and \(J_0\) and \(J_1\) are Bessel functions. The lines of force of such a field have the form of spirals of constant pitch1. In cases where the dissipation of the magnetic field due to the finite conductivity of the medium cannot be neglected, it can be shown [^102] that, under the condition \(\alpha=\mathrm{const}\), the magnetic field remains force-free in the process of dissipation, i.e., hydrostatic equilibrium is preserved.

For applications, solutions of equation (4.4) that decrease or pass into a uniform field at infinity are of greater interest. Under the condition \(\alpha=\mathrm{const}\) and assuming cylindrical symmetry of the problem, a particular solution of equations (4.2), (4.4) was found in [^106]. This solution corresponds to a field decomposing into separate spherical layers, inside each of which the lines of force are closed. At the boundary of a layer, matching of solutions with different \(\alpha\) is possible, as well as with a solution passing at infinity into a uniform field. Under the same assumptions, in [^42] an explicit general solution of equations (4.2), (4.4) was obtained, expressed through cylindrical functions of \(r\) and Gegenbauer polynomials in \(\cos\vartheta\) in spherical coordinates \(r,\vartheta,\varphi\). Both of these papers use the method of decomposing a cylindrically symmetric field into poloidal and toroidal parts, for the first of which the magnetic-field intensity vector \(\mathbf H\) lies in the plane passing through the axis of symmetry, while for the second it is perpendicular to it. Each of these parts is completely determined by one scalar function of the cylindrical coordinates \(\rho\) and \(z\). With the aid of this decomposition, in [^43] a general relation was obtained between the defining scalars of a cylindrically symmetric magnetic field satisfying equation (4.1), with the forces of gravity taken into account.

In [^51] a class of two-dimensional static solutions was found for a compressible isothermal medium in a uniform gravitational field. This class of solutions is determined by a family of lines of force possessing the property that an arbitrary magnetic field whose lines of force belong to this family can be in equilibrium with the medium for a corresponding pressure distribution. We note that in the absence of gravity such a family is the family of concentric circles. In this case the meaning of the solution is especially simple: for any magnetic field with such lines of force, there evidently exists a pressure distribution balancing the tensions of the magnetic field.

Among the problems of magnetohydrostatics there also belongs the question, important for the theory of magnetic stars, of the stable configuration of a gravitating liquid sphere in the presence of a magnetic field. In [^28] [^73] [^126] [^6] it was shown that, in the presence of a uniform internal and dipole external magnetic fields, the sphere is not an equilibrium configuration of a gravitating liquid mass. Under the influence of the additional “lateral” magnetic pressure, the sphere will turn into a spheroid compressed in the direction of the magnetic field. If the magnetic-field intensity exceeds a certain critical value, then a stable configuration in general

impossible: the magnetic field stretches the spheroid in the transverse directions. This occurs if the magnetic energy exceeds the gravitational energy of the star.

b) Stationary solutions

As follows from equations (1.35)—(1.39), the stationary motion of a conducting medium in a magnetic field is governed by the following equations:

\[ \operatorname{rot}[\mathbf{vH}]=0, \tag{4.6} \]

\[ (\mathbf{v}\nabla)\mathbf{v}-\frac{1}{4\pi\rho}(\mathbf{H}\nabla)\mathbf{H} = -\frac{1}{\rho}\nabla\left(p+\frac{H^2}{8\pi}\right), \tag{4.7} \]

\[ \operatorname{div}\rho\mathbf{v}=0, \tag{4.8} \]

\[ \operatorname{div}\mathbf{H}=0, \tag{4.9} \]

\[ \mathbf{v}\nabla s=0. \tag{4.10} \]

The induction equation (4.6), or the equation equivalent to it,

\[ [\mathbf{vH}]=\operatorname{grad}\varphi, \tag{4.11} \]

where \(\varphi\) is an arbitrary function of the coordinates, shows that, for the magnetic field to be stationary, it is necessary that the electric field

\[ \mathbf{E}=-\frac{1}{c_0}[\mathbf{vH}] \]

(see (3.2)) be irrotational. In this case \(\varphi\) differs from the potential of the electric field only by the constant factor \(c_0\). Condition (4.11) means that in the stationary case the vectors \(\mathbf{v}\) and \(\mathbf{H}\) are perpendicular to the gradient of the potential, i.e. the magnetic lines of force and the current lines of the fluid lie on equipotential surfaces of the electric field.

The induction equation (4.6) is, evidently, satisfied in the following particular cases \(^{106}\): when the medium moves along the lines of force of the magnetic field

\[ \mathbf{v}\parallel\mathbf{H}; \tag{4.12} \]

under rigid rotation about the axis of symmetry in a cylindrically symmetric magnetic field

\[ \mathbf{v}=[\mathbf{r}\boldsymbol{\omega}], \tag{4.13} \]

where the angular velocity \(\boldsymbol{\omega}\) is a constant vector; for a rotation in which \(\mathbf{H}\) and \(\boldsymbol{\omega}\) are cylindrically symmetric, with \(\boldsymbol{\omega}\) everywhere parallel to the axis of symmetry and the condition

\[ \mathbf{H}\nabla\omega=0 \tag{4.14} \]

is fulfilled.

The last case is called isorotation in the literature \(^{62,3}\). As follows from condition (4.14), under such rotation the lines of force of the magnetic field are situated on the surfaces \(\omega=\mathrm{const}\). Since these surfaces, owing to cylindrical symmetry, are surfaces of rotation, it is clear that under such motion the magnetic field remains unchanged. The cases of rigid rotation and isorotation are widely discussed in applications to stellar physics, since rotation is assumed to be characteristic of the majority of stars possessing strong magnetic fields.

The force-free fields discussed above, satisfying conditions (4.3) or (4.4), are possible not only in the static case, but also under

solid rotation in an incompressible or barotropic medium. \([\nabla p \cdot \nabla \rho]=0\), as follows from the remaining unused equations of motion of the medium (4.7) and (4.8). The question of the compatibility of the general case of isorotation with the equations of motion of the medium (4.7) and (4.8) has not been investigated.

In the case of an incompressible medium, the equations of magnetic hydrodynamics have an important class of stationary solutions of the form

\[ \mathbf{v}=\pm \frac{\mathbf{H}}{\sqrt{4\pi\rho}}, \tag{4.15} \]

where \(\mathbf{H}\) is an arbitrary magnetic field satisfying the usual condition

\[ \operatorname{div}\mathbf{H}=0. \tag{4.16} \]

It is easy to see that this solution satisfies all equations (4.6)—(4.9); moreover, the pressure \(p\) is related to the magnetic-field strength by the relation

\[ \nabla\left(p+\frac{H^2}{8\pi}\right)=0,\qquad p+\frac{H^2}{8\pi}=\mathrm{const}, \tag{4.17} \]

which follows from equation (4.7). In such motions, the density of the kinetic energy of the medium is everywhere equal to the density of the magnetic energy, and the Maxwell stresses of the magnetic field are exactly balanced by the hydrodynamic forces.

The existence of solutions (4.15) means that an arbitrary magnetic field and a moving conducting medium are in equilibrium if the motion of the medium occurs along the lines of force of this field with a velocity depending at each point on the magnetic-field strength according to expression (4.15). Stationary solutions of this type may be either continuous throughout space or may possess surfaces of discontinuity of the quantities \(p,\rho,\mathbf{v}\), and \(\mathbf{H}\). Let us note that, by virtue of incompressibility, a jump in density is possible only at the interface between two different media. As follows from Section 3, in an incompressible medium only two types of discontinuity surfaces are possible: a magnetohydrodynamic wave and a tangential discontinuity. The first of these is simply a particular case of solution (4.15), in which, instead of a smooth change, there is a sharp change in the direction of the magnetic-field lines. More interesting in connection with solution (4.15) is the case of a tangential-discontinuity surface. In this case the lines of force and the current lines of the fluid are parallel to the discontinuity surface. At the discontinuity surface, the velocity and the field strength may undergo an arbitrary jump, remaining connected by condition (4.15), while the total “lateral” pressure \(p+H^2/8\pi\), by virtue of equation (4.17), is continuous:

\[ p_1+\frac{H_1^2}{8\pi}=p_2+\frac{H_2^2}{8\pi}. \tag{4.18} \]

In particular, such a discontinuous solution may be realized in the form of a jet of arbitrary shape flowing in a stationary medium in which the magnetic field is absent, and separated from it by a surface of tangential discontinuity. Such jets may close into rings and loops of arbitrary form. As will be shown in the next paragraph, discontinuous solutions of this type are dynamically stable as a consequence of the stability of the tangential discontinuity with respect to small perturbations. Solution (4.15) was indicated in works \(^{141}\) and, independently, in work \(^{44}\).

In a compressible medium, solutions (4.15) are possible provided that, along the lines of force of the magnetic field, the density of the medium, the pressure, and the absolute

the values of the magnetic-field intensity remain constant\(^{141}\)*). This means that each tube of force of the magnetic field has a constant cross section, and the motion of the fluid along it occurs with constant velocity.

In investigations of magnetohydrodynamic motions under laboratory conditions, where the finite conductivity of the medium has a substantial effect (the practically attainable values of the number \(R_m\) are less than, or of the order of, unity), it is necessary to take into account the dissipation of the magnetic field. In this case a stationary process can occur only under the action of a constant external force, for example a pressure gradient.

In particular, a practically important case is that of stationary flow of a conducting medium through tubes in the presence of a transverse magnetic field. By measuring the potential difference of the induced electric field at different points of the tube cross section, one can judge the flow velocity and the total rate of liquid flow, which, for example, is important in working with a metallic heat-transfer agent in nuclear reactors \(^{134,135}\).

The problem of stationary one-dimensional flow of a conducting liquid between two parallel planes in the presence of a transverse external magnetic field was solved in work \(^{74}\). In this case the equations of magnetic hydrodynamics (1.19)—(1.22) reduce to the following (the \(z\)-axis is chosen in the direction of the uniform external magnetic field \(H_0\), the \(x\)-axis in the direction of the velocity):

\[ H_0 \frac{dv}{dz} + \nu_m \frac{\partial^2 h}{\partial z^2} = 0, \tag{4.19} \]

\[ \rho\nu \frac{\partial^2 v}{\partial z^2} + \frac{H_0}{4\pi}\frac{\partial h}{\partial z} - \frac{\partial p}{\partial x} = 0, \tag{4.20} \]

\[ \frac{\partial}{\partial z}\left(p+\frac{h^2}{8\pi}\right)=0. \tag{4.21} \]

In these equations \(H_0\) is the transverse component of the magnetic-field intensity, which, by virtue of the equation \(\operatorname{div}\mathbf H=0\), does not depend on \(z\) and coincides with the intensity of the external magnetic field, while \(h\) is the component of the field along the velocity. From equations (4.19) and (4.20) it follows that

\[ \frac{d^3 v}{dz^3}-\frac{H_0^2}{4\pi\rho\nu\nu_m}\frac{dv}{dz}=0. \tag{4.22} \]

The solution of this equation, which vanishes on the boundary surfaces \(z=\pm l\), has the form

\[ v=v_0\,\frac{\operatorname{ch} M-\operatorname{ch}\dfrac{Mz}{l}}{\operatorname{ch} M-1}, \tag{4.23} \]

where \(v_0\) is the velocity at the center of the stream, and \(M\) is the dimensionless combination characteristic of the given flow,

\[ M=\frac{lH_0}{\sqrt{4\pi\rho\nu\nu_m}} =\frac{lH_0}{c_0}\sqrt{\frac{\sigma}{\rho\nu}}. \tag{4.24} \]

The influence of the transverse magnetic field is manifested in the appearance of an additional resistance to the motion of the liquid and in a change of the velocity profile. The velocity profile for different values of \(M\) is shown in

*) See also \(^{44a}\).

Fig. 6. Instead of the usual parabolic velocity profile, in a transverse magnetic field there is realized a profile with a smoother central part and a sharper decrease of the velocity near the boundary surfaces. For large values of \(M\), practically the entire change of velocity takes place in the near-wall layer of thickness \(l/M\).

Steady flows in a transverse magnetic field in tubes of rectangular and circular cross sections were considered in works \(^{134,135}\). Experimental investigation of such flows was carried out in works \(^{75,114}\), in which mercury was used as the conducting liquid.

c) Magnetohydrodynamic waves of arbitrary amplitude

Alfvén \(^{1,2,5}\) found a nonstationary solution of the equations of magnetic hydrodynamics for an incompressible fluid in the form of a wave of arbitrary amplitude traveling along an initially uniform magnetic field \(H_0\) with velocity

\[ V=\pm \frac{H_0}{\sqrt{4\pi\rho}}. \]

Such waves were called magnetohydrodynamic. In their example, the distinctive character of the dynamics of a conducting medium in a magnetic field was first revealed.

Assuming in the equations of magnetic hydrodynamics (1.35)—(1.38) that \(\rho=\mathrm{const}\) and \(\mathbf H=\mathbf H_0+\mathbf h\), where \(\mathbf H_0=\{H_0,0,0\}\) is a given constant magnetic field, we have

\[ \frac{\partial \mathbf h}{\partial t} = H_0\frac{\partial \mathbf v}{\partial x} -(\mathbf v\nabla)\mathbf h +(\mathbf h\nabla)\mathbf v, \tag{4.25} \]

\[ \frac{\partial \mathbf v}{\partial t} = -\frac{1}{\rho}\nabla\left(p+\frac{(\mathbf H_0+\mathbf h)^2}{8\pi}\right) +\frac{1}{4\pi\rho}H_0\frac{\partial \mathbf h}{\partial x} -(\mathbf v\nabla)\mathbf v +\frac{1}{4\pi\rho}(\mathbf h\nabla)\mathbf h, \tag{4.26} \]

\[ \operatorname{div}\mathbf v=0,\qquad \operatorname{div}\mathbf H=0. \tag{4.27} \]

In the solution found by Alfvén, the velocity of the medium \(\mathbf v\) and the deviation of the magnetic field \(\mathbf h\) from uniform are related by the condition

\[ \mathbf v=\mp \frac{\mathbf h}{\sqrt{4\pi\rho}}. \tag{4.28} \]

Applying the operation \(\operatorname{div}\) to equation (26) and using relations (4.27) and (4.28), we obtain

\[ \nabla^2\left(p+\frac{(\mathbf H_0+\mathbf h)^2}{8\pi}\right)=0. \tag{4.29} \]

Since everywhere outside the wave \(\mathbf h=0\) and \(p=p_0\), it follows that

\[ p+\frac{(\mathbf H_0+\mathbf h)^2}{8\pi} = p_0+\frac{H_0^2}{8\pi} = \mathrm{const}. \tag{4.30} \]

This equation shows that in a magnetohydrodynamic wave the ordinary pressure \(p\) is everywhere balanced by the magnetic pressure \(H^2/8\pi\). At the same time, from the expression for the Lorentz force

\[ \mathbf f = \frac{1}{4\pi}[\operatorname{rot}\mathbf H\cdot \mathbf H] = \frac{1}{4\pi}(\mathbf H\nabla)\mathbf H - \frac{1}{8\pi}\nabla H^2 \tag{4.31} \]

it follows that the motion occurs under the action of its uncompensated part \(\frac{1}{4\pi}(\mathbf{H}\nabla)\mathbf{H}\), which corresponds to Maxwell stresses along the lines of force of the magnetic field. The latter circumstance makes it possible to draw an analogy between magnetohydrodynamic waves and transverse elastic waves in a string\(^5\).

Under the conditions (4.28) and (4.30), equations (4.25) and (4.26) reduce to the following:

\[ \left. \begin{aligned} \frac{\partial h}{\partial t} &= H_0 \frac{\partial v}{\partial x},\\ \frac{\partial v}{\partial t} &= \frac{H_0}{4\pi \rho}\frac{\partial h}{\partial x}. \end{aligned} \right\} \tag{4.32} \]

Hence follows the equation for the velocity

\[ \frac{\partial^2 v}{\partial t^2} - \frac{H_0^2}{4\pi \rho} \frac{\partial^2 v}{\partial x^2} =0 \tag{4.33} \]

and an analogous equation for \(\mathbf{h}\). Equation (4.33) is a one-dimensional wave equation. This means that an arbitrary initial velocity profile \(\mathbf{v}\) (and the magnetic-field intensity \(\mathbf{h}\) related to it by condition (4.28)) propagates along the constant initial field \(H_0\) with velocity

\[ V=\pm \frac{H_0}{\sqrt{4\pi \rho}}. \tag{4.34} \]

The distribution of the velocity and of the magnetic-field intensity in the wave must, of course, satisfy conditions (4.27). Magnetohydrodynamic waves in an incompressible fluid and, in particular, vortex rings propagating along the field are considered in detail in \({}^{145}\).

As is easy to see, the shock magnetohydrodynamic wave considered in Section 3 is a special case of the solution (4.28), (4.30), corresponding to a discontinuous velocity profile.

г) Nonstationary motions

In view of the mathematical complexity of the equations of magnetic hydrodynamics, the finding of exact nonstationary solutions encounters serious difficulties. At present, only the problem of one-dimensional motion of an ideal medium has been studied in detail\({}^{85,99}\). Apart from the trivial case of a magnetic field constant throughout all space, one-dimensional motion is possible only in a transverse magnetic field, since only under this condition is the Lorentz force directed along the velocity. Let the motion occur along the \(x\)-axis, let all quantities depend only on the coordinate \(x\) and time \(t\), and let the magnetic field be directed along the \(y\)-axis. Then equations (1.35)—(1.39) reduce to the following:

\[ \frac{\partial v}{\partial t} + v\frac{\partial v}{\partial x} = -\frac{1}{\rho} \frac{\partial}{\partial x} \left( p+\frac{H^2}{8\pi} \right), \tag{4.35} \]

\[ \frac{\partial \rho}{\partial t} + v\frac{\partial \rho}{\partial x} = -\rho\frac{\partial v}{\partial x}, \tag{4.36} \]

\[ \frac{\partial s}{\partial t} + v\frac{\partial s}{\partial x} = 0, \tag{4.37} \]

\[ \frac{\partial H}{\partial t} + v\frac{\partial H}{\partial x} = -H\frac{\partial v}{\partial x}. \tag{4.38} \]

The last three equations immediately lead to the first integral

\[ H/\rho=b(s), \tag{4.39} \]

expressing the “attachment” of the lines of force to the medium: the magnetic-field strength changes proportionally to the density. Expressing, with the aid of (4.39), \(H\) in terms of \(\rho\) and \(s\), one can reduce equations (4.35)—(4.37) to the ordinary hydrodynamic equations with a new equation of state:

\[ p_m(\rho,s)=p(\rho,s)+\frac{b^2(s)}{8\pi}\rho^2. \tag{4.40} \]

Solutions of the corresponding problem for a number of cases are given in \({}^{85}\). In particular, the velocity of propagation of small disturbances is expressed in the usual way:

\[ c_m^2=\left(\frac{\partial p_m}{\partial \rho}\right)_s = c^2+\frac{H^2}{4\pi\rho} \tag{4.41} \]

and coincides with that found by us earlier for magnetosonic waves propagating across the field (cf. equations (2.18), (3.17)). Thus, the problem of one-dimensional motion in a perpendicular field reduces to an ordinary hydrodynamic problem with the equation of state modified in the corresponding way, as we have already seen in the example of a perpendicular shock wave.

For this same problem, in \({}^{99}\) the method of characteristics is developed and a scheme for numerical solution is given.

Under the assumption of cylindrical symmetry of the problem, the equations of magnetohydrodynamics for an incompressible inviscid fluid with finite conductivity are substantially simplified in \({}^{39}\) by decomposing the solenoidal vectors into toroidal and poloidal parts. The equations obtained there are used in papers \({}^{40,116}\) to determine the rate of dissipation of the magnetic field in a moving medium.

In view of the difficulty of finding exact solutions, a number of problems of magnetohydrodynamics are solved in the linear approximation, by investigating the behavior of small deviations from some prescribed state. In connection with such a formulation of the problem, we note papers \({}^{131,72,63,46,123,47,90,41}\), in which magnetohydrodynamic oscillations of a gravitating liquid sphere are studied; these are of interest for stellar physics.

5. AMPLIFICATION OF THE MAGNETIC FIELD. THE HYDROMAGNETIC DYNAMO

One of the important problems in the physics of cosmic processes is the origin and maintenance, on the average at a constant level, of cosmic magnetic fields. In principle, this problem can be solved within the framework of magnetohydrodynamics. It follows from the equations of magnetohydrodynamics that the magnetic field must increase both in chaotic turbulent motion of a conducting medium and in certain regular motions. Turbulence in magnetohydrodynamics will be considered in Section 7. In the present paragraph we consider the process of amplification of the magnetic field under regular motions of a conducting medium. This range of questions belongs to the theory of the so-called hydromagnetic dynamo, which is being intensively developed at the present time as applied to the magnetic fields of the Earth and stars.

The process of amplification of the magnetic field in a hydromagnetic dynamo, as in an ordinary dynamo machine, consists in the fact that motions in a conducting medium lead to self-excitation and growth of a weak

of the initial magnetic field, the existence of which is not difficult to admit, by investigating the finer properties of the dynamics of an ionized medium\(^{130}\). In the presence of self-excitation, the magnetic field, despite dissipation, can be maintained at a constant level at the expense of the energy supplied by the hydrodynamic motion of the medium.

A rigorous solution of the problem of the behavior of a magnetic field frozen into a moving conducting medium must be based on the complete system of equations of magnetohydrodynamics. However, in view of the mathematical complexity of this path, which presupposes finding the general solution of the magnetohydrodynamic equations, it is practically hopeless. Therefore, at present the process of amplification of a magnetic field in a moving conducting medium is considered either semi-quantitatively, using the basic qualitative results of magnetohydrodynamics\(*\), or in a nonrigorous, purely kinematic formulation of the problem. Namely, some more or less reasonable state of motion of the medium is assumed to be given, and the behavior of the magnetic field associated with this medium is investigated. The nonrigor of such a formulation of the problem consists in the fact that the magnetic field has a back reaction on the motion of the medium, and therefore without an analysis of the complete system of magnetohydrodynamic equations one cannot be certain that the assumed hydrodynamic motion of the medium can actually take place.

A detailed review of works on the theory of the hydromagnetic dynamo is given in \(^{60}\) and \(^{58}\)\(**\). Below we shall only briefly discuss the initial premises and the main results of this theory.

The theory of the hydromagnetic dynamo is based entirely on the induction equation

\[ \frac{\partial \mathbf{H}}{\partial t}=\operatorname{rot}[\mathbf{v}\mathbf{H}]+\nu_m \Delta \mathbf{H}, \tag{5,1} \]

in which the velocity of the medium is assumed to be a given function of the coordinates (and of time). Multiplying equation (5,1) by \(\mathbf{H}\) and using the identity \(\operatorname{div}[\mathbf{a}\mathbf{b}]=\mathbf{b}\operatorname{rot}\mathbf{a}-\mathbf{a}\operatorname{rot}\mathbf{b}\), it is easy to obtain

\[ \frac{\partial}{\partial t}\left(\frac{H^2}{2}\right) = \mathbf{v}[\mathbf{H}\operatorname{rot}\mathbf{H}] - \nu_m(\operatorname{rot}\mathbf{H})^2 - \operatorname{div}\{[\mathbf{H}[\mathbf{v}\mathbf{H}]]-\nu_m[\mathbf{H}\operatorname{rot}\mathbf{H}]\}. \tag{5,2} \]

This equation is the differential form of the law of conservation of energy for the field. Integrating over some volume \(V\), bounded by a surface \(S\), we find

\[ \frac{\partial}{\partial t}\int_V \frac{H^2}{8\pi}\,dV = \frac{1}{4\pi}\int_V \mathbf{v}[\mathbf{H}\operatorname{rot}\mathbf{H}]\,dV - \frac{\nu_m}{4\pi}\int_V(\operatorname{rot}\mathbf{H})^2\,dV - \]

\[ -\frac{1}{4\pi}\int_S \{[\mathbf{H}[\mathbf{v}\mathbf{H}]]-\nu_m[\mathbf{H}\cdot \operatorname{rot}\mathbf{H}]\}\,dS . \tag{5,3} \]

Thus, the change in the energy of the magnetic field in a certain volume is determined by the work performed by the Lorentz force \(-\dfrac{1}{4\pi}[\mathbf{H}\operatorname{rot}\mathbf{H}]\) on the fluid, by the Joule losses \(\dfrac{\nu_m}{4\pi}(\operatorname{rot}\mathbf{H})^2=j^2/\sigma\), and by the flux of electromagnetic energy through the surface bounding the volume under consideration

* See, for example, \(^{49}\), where a rough model of field amplification in a conducting liquid sphere is given.

** See also \(^{23,60a,47a}\).

volume. Indeed, the expression under the sign of the surface integral is the electromagnetic part of the vector of energy flux \(\mathbf g\), defined by expression (1.28). If there is no field outside the volume under consideration, the surface integral in equation (5.3) vanishes.

The problem of the theory of the hydromagnetic dynamo consists in finding such a velocity field for which the first term on the right-hand side of equation (5.3) is greater than the second, or equal to it. This ensures the growth or stationarity of the energy of the magnetic field. As follows from equation (5.3), in such a velocity field the hydrodynamic forces must do work against the Lorentz force, on the average over the volume of the fluid. Below, the properties of such a velocity field will be considered in more detail.

For the time being we shall confine ourselves to purely induction effects, neglecting dissipation of the magnetic field. Setting \(\nu_m=0\) in equation (5.1) and transforming the first term on the right-hand side, we find

\[ \frac{\partial \mathbf H}{\partial t}+(\mathbf v\nabla)\mathbf H = (\mathbf H\nabla)\mathbf v-\mathbf H\,\operatorname{div}\mathbf v . \tag{5.4} \]

With the aid of the continuity equation (1.13), equation (4) is easily transformed into the following form[^144]:

\[ \frac{d}{dt}\left(\frac{\mathbf H}{\rho}\right) = \left(\frac{\mathbf H}{\rho}\nabla\right)\mathbf v, \tag{5.5} \]

where \(\dfrac{d}{dt}=\dfrac{\partial}{\partial t}+(\mathbf v\nabla)\) is the derivative with respect to time at a point moving with the medium. Equation (5.5) is, in form, analogous to Helmholtz’s equation for vorticity in ordinary hydrodynamics and has the general solution

\[ \frac{\mathbf H}{\rho} = \frac{\mathbf H_0}{\rho_0} + \left(\frac{\mathbf H_0}{\rho_0}\nabla_0\right)\boldsymbol{\xi}, \tag{5.6} \]

where \(\mathbf H\) and \(\rho\) are the magnetic-field strength and the density of the medium for a volume element which, in time \(t\), has undergone the displacement \(\boldsymbol{\xi}(\mathbf r_0,t)\) from the initial position \(\mathbf r_0\), in which the field strength and density were \(\mathbf H_0\) and \(\rho_0\). Differentiation of the vector \(\boldsymbol{\xi}\) is performed with respect to the coordinates of the initial point, which is denoted by the subscript 0.

Equation (5.6) makes it possible, for a prescribed motion of the fluid, to calculate the field strength at any instant of time, if the initial field strength is known. Many results, however, can be obtained directly from theorem (1.45) on the conservation of magnetic flux through an arbitrary material surface.

It is easy to see that there is a whole series of motions of the fluid for which the field strength increases[^50]. Thus, for example, under uniform compression of the medium the cross section of each fluid particle decreases by a factor \((\rho/\rho_0)^{2/3}\), and from conservation of magnetic flux it follows that the field strength grows proportionally to \(\rho^{2/3}\). More interesting are motions of an incompressible fluid, or motions in which the fluid behaves, on the average over a long time, as incompressible. In this case equation (5.6) reduces to the following:

\[ \mathbf H=\mathbf H_0+(\mathbf H_0\nabla_0)\boldsymbol{\xi}. \tag{5.7} \]

From equation (5.7), or directly from the theorem on the conservation of magnetic flux through a material surface, it follows that any motion of the medium in which fluid points situated on one and the same line of force move away from one another leads to an increase of the magnetic-field strength proportional to the distance between

these points. Indeed, elongation of a fluid particle by a factor \(l\) along the magnetic field means, with unchanged volume, a reduction by a factor \(l\) of its transverse cross-section. By virtue of the invariance of the flux through the transverse section, this means that the field strength also increases by a factor \(l\), i.e. it grows proportionally to the elongation of an element of the medium in the direction of the field. Thus, field growth is produced by such motions as “stretch” the magnetic-field lines frozen into the medium.

If the velocity distribution does not depend on time and the conductor is ideal \((\nu_m=0)\), then a stationary state, generally speaking, cannot be reached, since the magnetic field will grow without bound \(^{15}\). Therefore, in the theory of the stationary hydromagnetic dynamo it is necessary to take into account the dissipation of the magnetic field. Moreover, for any theory of the hydromagnetic dynamo, what is important is not only field growth but also the smoothing of its inhomogeneities, as a result of which the initial more or less homogeneous field is regenerated. This also makes it necessary to consider the dissipation of the magnetic field. If the conductivity of the medium is finite, then for a given velocity field and initial magnetic field, as a result of dissipation of the magnetic field, some stationary state is always established. A number of simple examples are considered in \(^{4}\).

For a stationary process, equation (5.1) is equivalent to the following:

\[ [\mathbf{v}\mathbf{H}]=\nu_m \operatorname{rot}\mathbf{H}+\nabla \varphi, \tag{5.8} \]

where the arbitrary function \(\varphi\), as is easily seen from expression (1.8), differs only by a constant factor \(c_0\) from the potential of the electrostatic field of the charges. Points at which \(\mathbf{H}=0\) are special for any induction mechanism of amplification of the magnetic field. At these points the induction mechanism does not act, and the current

\[ \mathbf{j}=\frac{c_0}{4\pi}\operatorname{rot}\mathbf{H} \]

can be due only to the electrostatic field of the charges. If the latter is absent (in a number of cases this is required by considerations of spatial symmetry), then at points where \(\mathbf{H}=0\), there will also be \(\mathbf{j}=0\) and \(\operatorname{rot}\mathbf{H}=0\). The latter condition imposes definite restrictions on the properties of the stationary field in the neighborhood of a zero point. Namely, the lines of force in the neighborhood of such a point cannot be closed, since otherwise the integral \(\oint \mathbf{H}\,dl=\int \operatorname{rot}\mathbf{H}\,ds\), taken along a line of force in the neighborhood of the zero point, would differ from zero, which is impossible when \(\operatorname{rot}\mathbf{H}=0\).

From the condition \(\operatorname{rot}\mathbf{H}=0\) at points where \(\mathbf{H}=0\), there immediately follows the impossibility of a stationary hydromagnetic dynamo in a two-dimensional magnetic field vanishing at infinity. By a two-dimensional magnetic field one generally means a field whose lines of force lie on arbitrary surfaces that divide space into simply connected regions and, consequently, by virtue of the condition \(\operatorname{div}\mathbf{H}=0\), either are closed or end at infinity. If the magnetic field vanishes at infinity, then all its lines of force are closed and, consequently, there exists at least one point at which \(\mathbf{H}=0\) and which is encircled by lines of force. As we have already seen, in the stationary problem, in the absence of currents caused by the electrostatic field of charges, the magnetic field must vanish also in the neighborhood of such a point. Continuing analogous arguments, we arrive at the conclusion that the magnetic field must be absent throughout all space. Thus, in a two-dimensional magnetic field, in the absence of currents caused by the electrostatic field of charges, a stationary dynamo is impossible. The dissipation of the magnetic

MAGNETIC HYDRODYNAMICS

the field at points at which \(\mathbf{H}=0\) is not compensated by the induction mechanism, since the latter does not act at these points, and, consequently, the magnetic field decays, contracting toward these points.

This theorem was first established by Cowling \(^{45}\) for the cylindrically symmetric case, under the condition that the vectors of the magnetic field and the velocity are situated in planes passing through the axis of symmetry. The absence of an electrostatic field in this case follows directly from symmetry considerations. In the review \(^{58}\) it is asserted that the restriction adopted by Cowling, requiring that the velocity of the medium lie in planes passing through the axis of symmetry, is not essential. This assertion is valid only in the case of the absence of space charges induced by the motion of the fluid, which the author does not consider. However, the absence of such charges for an arbitrary magnetohydrodynamic motion is not obvious. On the contrary, equation (1.10) shows that in the general case such charges exist.

With the aid of the above-stated generalization of Cowling’s theorem it is not difficult to verify that a stationary dynamo is also impossible for an arbitrary plane two-dimensional motion. For this it is sufficient to show that, for such a motion, an electrostatic field cannot be maintained and, consequently, the magnetic field decays at points where \(\mathbf{H}=0\). Let the vectors \(\mathbf{v}\) and \(\mathbf{H}\) lie in the plane \((x,y)\) and not depend on \(z\). In this case, according to (1.9),

\[ \rho_e=-\frac{\varepsilon_0}{4\pi c_0}\operatorname{div}[\mathbf{v}\mathbf{H}]=0, \tag{5,9} \]

since \([\mathbf{v}\mathbf{H}]\) has a component only in the direction of the \(z\)-axis and does not depend on \(z\). This means that charges, and consequently also the electrostatic field, are absent throughout all space. Thus a two-dimensional hydromagnetic dynamo proves to be impossible.

The results presented show that a stationary dynamo process is impossible under high spatial symmetry of the magnetic field and the velocity field, and make quite plausible the assertion that a hydromagnetic dynamo must be essentially three-dimensional \(^{58}\). We emphasize, however, that this assertion has not been rigorously proved, since it has not been proved that, in the general case of a two-dimensional magnetic field considered above, the current at the zero points cannot be maintained by the electrostatic field of charges arising in magnetohydrodynamic motion.

As for a three-dimensional dynamo, its possibility can hardly be doubted, if only by analogy with ordinary technical generators made of solid conductors. Let us note that topologically possible bounded three-dimensional magnetic fields without zero points are possible; therefore the difficulties encountered in connection with the presence of such points in two-dimensional problems are absent here.

The works on the theory of the hydromagnetic dynamo available at present are connected mainly with the theory of terrestrial and stellar magnetism. We shall mention first of all the works \(^{53,54,21}\) and especially \(^{22}\), in which the stationary induction equation (5.8) is investigated for a prescribed velocity field, more or less corresponding to the assumed motions inside the star. Since in all these works a bounded mass of fluid having the shape of a sphere is considered, the expansion of the vector fields \(\mathbf{v}\) and \(\mathbf{H}\) in vector orthogonal spherical harmonics is used. In agreement with the analysis set forth above, it is shown that, under high symmetry of the vector fields, a stationary process is impossible. Therefore

spherical harmonics of higher orders, which break the symmetry of the velocity field, are included in the consideration. In paper ²² the resulting system of equations is solved numerically for several first terms of the expansion of the field. Unfortunately, the existence of a stationary hydromagnetic dynamo for the adopted velocity field remains unproved, since the solution is obtained formally in the form of a series whose convergence has not been clarified.

A different approach to the problem of maintaining the magnetic field of stars on the average at a stationary level is adopted in paper ¹¹⁸ and in the supplementary paper ¹¹⁹. These works use semi-quantitative considerations based on examining specific conditions in the metallic core of the Earth and in stars. As was shown in ²⁰ ²¹, the nonuniform rotation of stars, which in itself can be explained by convective motions that equalize the angular momentum at all points of the rotating star, should lead to the formation, from an initial dipole field, of a toroidal field (whose lines of force encircle the axis of rotation of the star). This process is possible only in one direction: the reverse process of forming a dipole field from a toroidal one, with a cylindrically symmetric distribution of velocity, does not occur. For the regeneration of a dipole field there must exist a certain asymmetry of the motion. The most natural cause of such asymmetry should be considered to be radial convective flows. As a result of the action of the Coriolis force these flows have the character of cyclones, which deform the lines of force of the toroidal field and form from them loops contributing to the initial dipole field. This scheme for the regeneration of the dipole field in a nonuniformly rotating conducting liquid sphere containing a convective zone has been considered in greatest detail by Parker ¹¹⁸, ¹¹⁹ᵃ.

6. PROBLEMS OF STABILITY

a) Gravitational stability

Questions of the stability of various configurations of a gravitating medium are very important for astronomy and astrophysics. Their consideration is necessary in the theories of stars and nebulae, and in the theory of the structure of the Galaxy, in which gravitation and, apparently, the magnetic field play a primary role. Below, the principal results obtained in this area are briefly presented.

It is known that a homogeneous distribution, unlimited in space, of a gravitating gas is unstable with respect to perturbations with wavelength

\[ \lambda^2 > \lambda_c^2 = \frac{\pi c^2}{G \rho}, \tag{6,1} \]

where \(c\) is the speed of sound, \(G\) the gravitational constant. When this condition, usually called the Jeans criterion, is fulfilled, a homogeneous distribution of gas under the action of gravitation breaks up into condensations of size of order \(\lambda_c\). The Jeans criterion also remains unchanged in the presence of a homogeneous magnetic field ²⁸. This result is obvious, since the magnetic field has no influence on displacements of the medium along the field. In paper ³¹ it is shown that the Jeans criterion is preserved also under the simultaneous action of a magnetic field and Coriolis forces arising as a result of uniform rotation of the medium.

The stability of an infinite cylinder formed by a conducting medium in the presence of a magnetic field parallel to the axis of the cylinder was considered in paper ²⁸. This problem is an idealization of conditions in the spiral arms of the Galaxy and makes it possible to estimate the upper limit

the intensity of the galactic magnetic field. With the aid of a generalization, obtained in this work, of the virial theorem that includes allowance for the magnetic field, the necessary condition for stability was found:

\[ \sqrt{\overline{H^2}} < 2\pi G\rho R, \tag{6.2} \]

where \(\overline{H^2}\) is the mean square of the magnetic-field intensity in the cylinder, \(R\) is the radius of the cylinder; a method was also indicated for determining the period of small adiabatic radial pulsations of the cylinder. The periods of such pulsations were calculated in work \(^{108}\). Condition (6.2) leads to the value \(6\cdot 10^{-6}\) oersted for the magnetic-field intensity in the spiral arms.

The stability of an infinite homogeneous cylinder of incompressible fluid with respect to perturbations in which the radius changes periodically according to the law

\[ r = R + a\cos kz, \tag{6.3} \]

where \(z\) is the coordinate along the axis of the cylinder, \(a\) is the small amplitude of the perturbation \((a \ll R)\), and \(k\) is the wave number of the perturbation, was investigated in \(^{28*}\). It was shown that the critical wavelength of the perturbation at which the cylindrical distribution of matter becomes unstable, and the wavelength \(\lambda_m\) for which the instability is maximal (the growth time of the instability is minimal), increase monotonically with increasing magnetic-field intensity. Applying the results obtained to the spiral arms of the Galaxy, the authors found that, in order for the growth time of the instability to be at least \(5\cdot 10^9\) years (the age of the Galaxy), the magnetic field must be of the order of \(7\cdot 10^{-6}\) oersted. This value is consistent with other estimates of the interstellar magnetic field.

For a gas of finite mass situated in a magnetic field, work \(^{28}\) obtained the following generalization of the virial theorem:

\[ \frac{1}{2}\frac{d I^2}{dt^2}=2T+3(\gamma-1)U+\mathfrak{M}+\Omega, \tag{6.4} \]

where \(I\) is the moment of inertia of the system; \(T\), \(U\), and \(\Omega\) are the kinetic, thermal, and gravitational energies of the gas; \(\gamma\) is the ratio of specific heats at constant pressure and constant volume. Relation (6.4) differs from the usual virial theorem by the presence of the term \(\mathfrak{M}=\dfrac{1}{8\pi}\iiint H^2 dV\), expressing the magnetic energy of the system under consideration. For a gas in equilibrium, equation (6.4) gives:

\[ 3(\gamma-1)U+\mathfrak{M}+\Omega=0. \tag{6.5} \]

With the aid of this relation, the total energy of the system \(\varepsilon=U+\mathfrak{M}+\Omega\) may be expressed as follows:

\[ \varepsilon=-\frac{3\gamma-4}{3(\gamma-1)}\left(|\Omega|-\mathfrak{M}\right). \tag{6.6} \]

Therefore the necessary condition for dynamical stability of a system in equilibrium, \(\varepsilon<0\), is equivalent, for \(\gamma>4/3\), to the condition

\[ |\Omega|-\mathfrak{M}>0. \tag{6.7} \]

This condition has a simple meaning: for a sufficiently strong magnetic

\[ \text{*) The article has been translated in collection }^{109}. \]

the gravitational field cannot balance the magnetic expansion of parts of the system. Condition (6.7) makes it possible to estimate the upper limit of the mean magnetic-field strength of stars. Substituting into (6.7), as \(\Omega\), the gravitational energy of a homogeneous sphere, we find

\[ \sqrt{\overline{H^2}} < 2\cdot 10^8\,\frac{M}{R^2}, \tag{6.8} \]

where the mass of the star \(M\) and its radius \(R\) are expressed in solar units. If the magnetic field inside the star is homogeneous, then as the limit established by condition (6.8) is approached, the shape of the star differs more and more from spherical: the equilibrium configuration is, as was already indicated above, a spheroid compressed in the direction of the field \(^{28,73,126}\). If condition (6.8) is not fulfilled, the star, under the action of the lateral pressure of the magnetic field, assumes the form of a flat disk and disintegrates. In this respect the action of the magnetic field is analogous to the action of rotation \(^{65}\).

With the aid of the virial theorem (6.4), in \(^{32}\) an expression was obtained for the frequency \(\sigma\) of small adiabatic radial pulsations of a gaseous star in the presence of an internal magnetic field

\[ \sigma^2=\frac{(3\gamma-4)(|\Omega|-\mathfrak{M})}{I}. \tag{6.9} \]

It is obvious that the pulsation frequency may be arbitrarily small when \(\mathfrak{M}\to|\Omega|\), i.e., when the magnetic-field strength approaches the limit beyond which the star becomes unstable. Using this result, the authors interpret the slow variations of the magnetic field of magnetic variable stars as adiabatic pulsations of stars whose magnetic field is close to the limiting one. The observed period of such variations is equal to several days, which exceeds by at least an order of magnitude the period of adiabatic pulsations calculated without taking the magnetic field into account. At the same time it is known that for some magnetic variable stars the magnetic-field strength at the surface of the star is only about an order of magnitude smaller than the limiting value determined by condition (6.8). Therefore the mean field strength may be very close to the limiting value, and the period of adiabatic pulsations may be considerably greater than that calculated without taking the magnetic field into account \(^{27}\).

b) Thermoconvection in a magnetic field

The problem of the stability of static states of a conducting medium in a magnetic field also includes the problem of convective instability of a layer of liquid heated from below. In ordinary hydrodynamics thermoconvection has been well studied both theoretically and experimentally. The principal results pertaining to this are as follows. The stability of a layer of liquid with respect to thermoconvection is characterized by the dimensionless Rayleigh number

\[ \mathrm{Re}=\frac{g\alpha|\beta|}{\varkappa\nu}\,d^4, \tag{6.10} \]

in which \(g\) is the acceleration of gravity, \(\alpha\) is the coefficient of volume expansion, \(\beta\) is the temperature gradient (directed opposite to the force of gravity), \(\varkappa\) is the coefficient of thermal diffusivity, \(\nu\) is the kinematic viscosity, and \(d\) is the depth of the liquid layer. Thermoconvection sets in when the dimensionless parameter \(\mathrm{Re}\) reaches a certain definite (depending on the boundary conditions) value \(\mathrm{Re}_{\mathrm{cr}}\), equal in order of magnitude

quantities \(10^3\). When stationary thermoconvection sets in, the horizontal layer of liquid breaks up into separate cells, in each of which the motion proceeds in the same way (Bénard cells).

Investigations of the analogous problem in magnetohydrodynamics \(^{143,26,33}\) show that the magnetic field substantially hinders the onset of thermoconvection. A detailed consideration of thermoconvection in a viscous liquid of finite conductivity in the presence of a magnetic field \(^{26,33}\) shows that the critical Rayleigh number at which thermoconvection arises is a monotonically increasing function of the dimensionless parameter

\[ Q=\frac{H_1^2}{4\pi\rho}\,\frac{d^3}{\nu\gamma_m}, \tag{6,11} \]

where \(H_1\) is the vertical component of the magnetic-field intensity. Hence it follows that the higher the conductivity of the medium and the greater the vertical component of the magnetic field, the more difficult the onset of thermoconvection becomes. The dependence between \(\mathrm{Re}_{\mathrm{cr}}\) and \(Q\) was determined in work \(^{26}\) for three types of boundary conditions: a) both boundary surfaces of the liquid layer are free; b) the layer is bounded by two solid surfaces; c) on one side the liquid layer is bounded by a solid surface, and on the other by a free one. The corresponding curves are given in Fig. 7. At large \(Q\) the dependence of \(\mathrm{Re}_{\mathrm{cr}}\) on \(Q\) becomes linear.

Fig. 7.

If the magnetic field is directed vertically, then thermoconvection has the usual cellular structure. If, however, there is a horizontal component of the magnetic field, then at the stability boundary thermoconvection arises in the form of “rolls” with axes along the horizontal component of the field. In both cases \(Q\), and consequently also the critical Rayleigh number, depend only on the vertical component of the magnetic field.

The experimental study of thermoconvection in mercury placed in a magnetic field with an intensity of about 1500 oersted \(^{115}\) showed good agreement with Chandrasekhar’s theory \(^{26}\).

Theoretical consideration leads to the conclusion that convective instability may, generally speaking, arise both in the form of stationary convection and in the form of oscillations of increasing amplitude. The latter case is also called “overstability.” This term belongs to Eddington and has the following meaning. When a system deviates from a stable state, forces arise that return the system to its initial state. In “overstability” these forces are excessively large: they not only return the system to its initial state, but also cause a deviation of the opposite sign with an amplitude greater than the initial one, i.e. they lead to the appearance of oscillations of increasing amplitude. The two possibilities indicated follow simply from the fact that, for the assumed dependence

disturbations in time in the form \(e^{-i\omega t}\), the transition from stable states, to which \(\operatorname{Im}(\omega)<0\) corresponds (the disturbance decays with time), to unstable ones, for which \(\operatorname{Im}(\omega)>0\) (the disturbance grows with time), is possible either through the value \(\omega=0\), or, in the general case, through a value \(\omega\ne 0\), but \(\operatorname{Im}(\omega)=0\). The first case corresponds to a stationary disturbance, the second to oscillatory motion at the stability boundary.

As is known, in the absence of a magnetic field thermoconvection arises, upon reaching \(\operatorname{Re}_{\mathrm{cr}}\), in the form of stationary cellular convection. The occurrence of instability in the form of oscillations of increasing amplitude, or, in other words, overstability, proves to be impossible. (However, as Chandrasekhar has shown\({}^{30,35}\) and as was experimentally confirmed in work\({}^{66}\), instability of this kind is possible in a rotating fluid.) In magnetic hydrodynamics the situation is different\({}^{143,26*}\). For \(\nu_m>\chi\), instability can arise only in the form of stationary convection. This condition is satisfied with a large margin in laboratory experiments in magnetic hydrodynamics. For example, for mercury \(\nu_m=7.5\cdot 10^3\ \mathrm{cm}^2/\mathrm{sec}\), \(\chi=4.7\cdot 10^{-2}\ \mathrm{cm}^2/\mathrm{sec}\). In astrophysics, however, as a rule, \(\chi \gg \nu_m\). Under this condition instability arises in the form of stationary convection upon reaching \(\operatorname{Re}_{\mathrm{cr}}\), if \(Q < 27\pi^2\nu_m/\nu\), or in the form of oscillations of increasing amplitude at \(R > 27\pi^4\nu_m/4\nu\), if \(Q > 27\pi^2\nu_m/\nu\). In the latter case the frequency of oscillations at the stability boundary is equal to
\[ \omega=\frac{\pi}{d}\sqrt{\frac{H^2}{4\pi\rho}} . \]

In connection with the problem of thermoconvection, we point out works\({}^{34,38}\), in which the influence on convective instability of a magnetic field and rotation acting simultaneously was studied theoretically. Each of these factors by itself hinders the onset of thermoconvection; their simultaneous action leads to a complicated dependence of \(\operatorname{Re}_{\mathrm{cr}}\) on \(Q\) and on the dimensionless Taylor number characterizing the rotation,
\[ T=\frac{4\Omega^2}{\nu^2}\,d^4, \tag{6,12} \]
where \(\Omega\) is the vertical component of the angular velocity of the medium.

The question of the onset of thermoconvection is very important for applications to atmospheric physics and astrophysics. In this connection it should be noted that, for the ionosphere and the upper layers of the solar atmosphere, the results set forth above, obtained under the assumption of isotropy of conductivity and thermal conductivity, prove to be unsuitable\({}^{67}\). Here the dependence of the conductivity and thermal conductivity on the magnitude and direction of the magnetic field becomes essential. The corresponding refinement of the theory is contained in work\({}^{68}\).

c) Stability of the simplest flows in a magnetic field

The peculiarity of magnetic hydrodynamics is also manifested in the investigation of the stability of flows of a conducting medium in a magnetic field. Many motions which, in the absence of a magnetic field, are unstable and lead to the occurrence of turbulence, turn out to be stable—

* See also\({}^{142}\).

MAGNETIC HYDRODYNAMICS

...stable in a sufficiently strong magnetic field. The results of the investigations available at the present time make it possible to draw the following general conclusion about the influence of a magnetic field on the stability of hydrodynamic motions of a conducting medium: a magnetic field can only increase the stability of a given velocity distribution in comparison with the same velocity distribution in the absence of a magnetic field. This conclusion becomes physically evident if one takes into account that any turbulization of the flow, owing to the “frozen-in” character of the magnetic field, must be accompanied (see Section 5) by an increase in magnetic energy, which impedes such turbulization. We note that cases are possible in which the magnetic field substantially changes the initial velocity distribution in the flow, as a result of which the motion becomes unstable, as, for example, was observed in experiments with mercury in works^91,92,96^. However, such a changed velocity distribution, in accordance with the assertion made above, is unstable even in the absence of a magnetic field.

The simplest problem that makes it possible to obtain a rigorous and sufficiently general result is the problem of the stability of a tangential discontinuity^140^. Let us consider this problem for an incompressible liquid. Let the surface of the tangential discontinuity, coinciding with the plane \(x=0\), be subjected to a small perturbation, as a result of which \(x=\operatorname{Re}(\xi e^{i\mathbf{k}_0\mathbf{r}})\), where \(\mathbf{k}_0=\{0,k_y,k_z\}\) is a real vector, and \(\xi\) is the amplitude of the perturbation, small in comparison with the wavelength of the perturbation \(2\pi/k_0\). An arbitrary perturbation of the discontinuity surface can evidently be represented as a superposition of perturbations of this kind. Since the perturbation is assumed small, in each of the half-spaces \(x>0\) and \(x<0\), separated by the discontinuity surface, one may use the linearized system of magnetohydrodynamic equations (2,3). Solutions of this system are sought whose dependence on the coordinates and time has the form

\[ e^{i(\mathbf{k}\mathbf{r}-\omega t)}, \tag{6,13} \]

where \(\mathbf{k}=(k_x,k_y,k_z)\). Solutions of this form must satisfy equations (2,6), which for an incompressible liquid reduce to the following equations:

\[ (\omega-\mathbf{k}_0\mathbf{v})\mathbf{u}' + (\mathbf{k}_0\mathbf{u})\mathbf{v}'=0, \tag{6,14} \]

\[ (\omega-\mathbf{k}_0\mathbf{v})\mathbf{v}' + (\mathbf{k}_0\mathbf{u})\mathbf{u}'-\frac{1}{\rho}(p' + \rho \mathbf{u}\mathbf{u}')\mathbf{k}=0, \tag{6,15} \]

\[ (\mathbf{k}\mathbf{u}')=0,\qquad (\mathbf{k}\mathbf{v}')=0. \tag{6,16} \]

Here it has been taken into account that in a tangential discontinuity \(v_x=0\) and \(u_x=0\), and therefore \((\mathbf{k}\mathbf{v})=(\mathbf{k}_0\mathbf{v})\) and \((\mathbf{k}\mathbf{u})=(\mathbf{k}_0\mathbf{u})\). Multiplying equation (6,15) scalarly by \(\mathbf{k}\), we obtain

\[ k^2(p' + \rho\mathbf{u}\mathbf{u}')=0. \tag{6,17} \]

It follows from this that either \(k^2=0\), or \(p' + \rho\mathbf{u}\mathbf{u}'=0\). In the latter case, from equations (6,14) and (6,15) it follows that

\[ (\omega-\mathbf{k}_0\mathbf{v})^2-(\mathbf{k}_0\mathbf{u})^2=0. \tag{6,18} \]

Since this equation determines real values of \(\omega\), whereas instability can be associated only with the complex character of \(\omega\), we shall assume

\[ k^2 \equiv k_x^2+k_0^2=0,\qquad k_x=\pm i k_0. \tag{6,19} \]

The sign of \(k_x\) is chosen in accordance with the requirement that the disturbance be bounded far from the discontinuity: minus in the region \(x<0\) and plus in the region \(x>0\). Quantities pertaining to these regions will henceforth be denoted by the subscripts 1 and 2, respectively.

On the surface of discontinuity, the solutions of equations (6.14)—(6.16), taken for both regions, are joined by means of the following boundary equations at \(x=0\):

\[ \left. \begin{aligned} u'_{1x}-i(k_0u_1)\xi&=0,\\ u'_{2x}-i(k_0u_2)\xi&=0, \end{aligned} \right\} \qquad (6.20) \]

\[ p'_1+\rho_1\mathbf{u}_1\mathbf{u}'_1 = p'_2+\rho_2\mathbf{u}_2\mathbf{u}'_2 . \]

These conditions express the absence of normal components of the magnetic-field intensity and the continuity of the “total” pressure on the perturbed surface of the tangential discontinuity. Eliminating from equations (6.14)—(6.16), (6.19), and (6.20) the quantities \(\mathbf{u}'\), \(\mathbf{v}'\), \(p'\), \(k_x\), and \(\xi\), we obtain the following equation for \(\omega\):

\[ \rho_1(\omega-\mathbf{k}_0\mathbf{v}_1)^2 + \rho_2(\omega-\mathbf{k}_0\mathbf{v}_2)^2 - \rho_1(\mathbf{k}_0\mathbf{u}_1)^2 - \rho_2(\mathbf{k}_0\mathbf{u}_2)^2 =0 . \qquad (6.21) \]

The equation (6.21), quadratic with respect to \(\omega\), has two complex-conjugate roots under the condition

\[ \rho_1(\mathbf{k}_0\mathbf{u}_1)^2 + \rho_2(\mathbf{k}_0\mathbf{u}_2)^2 - \frac{\rho_1\rho_2}{\rho_1+\rho_2} \{\mathbf{k}_0(\mathbf{v}_2-\mathbf{v}_1)\}^2 <0 \qquad (6.22) \]

and two real roots in the opposite case. As is seen from expression (6.13), the presence of a complex root \(\omega\) with positive imaginary part means an unbounded growth of the disturbance with time and, consequently, instability of the tangential discontinuity. Thus, taking into account that \(\mathbf{u}=\dfrac{\mathbf{H}}{\sqrt{4\pi\rho}}\), we find that the stability condition for a tangential discontinuity in an incompressible medium has the form

\[ (\mathbf{k}_0\mathbf{H}_1)^2 + (\mathbf{k}_0\mathbf{H}_2)^2 - \frac{4\pi\rho_1\rho_2}{\rho_1+\rho_2} (\mathbf{k}_0\mathbf{v}_0)^2 \geqslant 0, \qquad (6.23) \]

where \(\mathbf{H}_1,\rho_1\) and \(\mathbf{H}_2,\rho_2\) are the magnetic-field intensity and the density of the medium on the two sides of the discontinuity surface, \(\mathbf{v}_0=\mathbf{v}_2-\mathbf{v}_1\) is the jump of velocity at the discontinuity, and \(\mathbf{k}_0\) is the wave vector of the disturbance, characterizing the direction of the “ripple” on the discontinuity surface. As follows from condition (6.23), the magnetic field makes a positive contribution to the left-hand side of the inequality and, consequently, exerts a stabilizing effect on the flow. Condition (6.23) does not depend on the absolute magnitude of the wave vector \(\mathbf{k}_0\), i.e., on the wavelength of the disturbance, but depends essentially on its direction. For example, the discontinuity is always stable with respect to disturbances for which \(\mathbf{k}_0\mathbf{v}_0=0\), i.e., the “ripple” on the discontinuity surface is situated along the relative velocity \(\mathbf{v}_0\). It also follows from condition (6.23) that the stabilizing effect is exerted only by the components of the magnetic field in the direction of the vector \(\mathbf{k}_0\). This was to be expected, since only these components undergo “stretching” when the surface of discontinuity is deformed.

In the general case of nonparallel \(\mathbf{H}_1,\mathbf{H}_2\), and \(\mathbf{v}_0\), the most “dangerous” disturbance, and consequently also the condition of stability of a tangential discontinuity with respect to arbitrary small disturbances, are determined

by the minimum of the left-hand side of expression (6.23) over all possible values of the vector \(\mathbf{k}_0\). In a coordinate system whose \(y\)-axis is directed along the jump of the velocity \(\mathbf{v}_0\), this condition has the form

\[ \frac{(H_{1y}H_{2z}-H_{2y}H_{1z})^2}{H_{1z}^2+H_{2z}^2} - \frac{4\pi \rho_1\rho_2}{\rho_1+\rho_2}\,v_0^2 \geqslant 0 . \tag{6.24} \]

If, however, the magnetic field on both sides of the discontinuity is directed along the velocity jump, then condition (6.23) is altogether independent of \(\mathbf{k}_0\), and, consequently, the discontinuity is stable with respect to arbitrary small perturbations if

\[ H_1^2+H_2^2-\frac{4\pi \rho_1\rho_2}{\rho_1+\rho_2}\,v_0^2 \geqslant 0 . \tag{6.25} \]

The latter case is the most interesting, since the motion of a medium along a magnetic field, as we have already seen, can be stationary also in a nonuniform magnetic field and velocity field. In particular, if the density of the medium and the magnetic-field strength are continuous, i.e. only the tangential component of the velocity undergoes a discontinuity, then condition (6.25) reduces to the following:

\[ \frac{H^2}{8\pi} \geqslant \frac{1}{4}\,\frac{\rho v_0^2}{2}. \tag{6.26} \]

This means that the discontinuity becomes stable with respect to small perturbations if the magnetic-energy density reaches \(1/4\) of the kinetic-energy density of the relative motion of the medium at the discontinuity.

As shown in \({}^{141}\), the stability condition for a tangential discontinuity in a compressible medium in the presence of a longitudinal magnetic field coincides with condition (6.26) everywhere except in the region of near-sonic velocities, where it differs only slightly from the latter.

In Section 4, exact stationary solutions of the equations of magnetohydrodynamics were considered, in which the velocity and the magnetic-field strength are related by

\[ \mathbf{v}=\pm \frac{\mathbf{H}}{\sqrt{4\pi\rho}} . \tag{6.27} \]

As was noted, such solutions may be realized, in particular, in the form of jets, loops, or analogous flows bounded by the surface of a tangential discontinuity. We shall now show that such flows are stable as a consequence of the stability of the surface of the tangential discontinuity. Using (6.27) in the general stability condition (6.23), we obtain the following stability condition for discontinuous stationary solutions obeying relation (6.27):

\[ \left\{\sqrt{\rho_1}\,(\mathbf{k}_0\mathbf{H}_1)\pm \sqrt{\rho_2}\,(\mathbf{k}_0\mathbf{H}_2)\right\}^2 \geqslant 0 . \tag{6.28} \]

This condition is, obviously, always satisfied. The stability of continuous solutions of the form (6.27) was considered in work \({}^{44}\).

The investigation of flows more complicated than a tangential discontinuity is very cumbersome and, generally speaking, can be carried out only by numerical methods. However, some qualitative consequences can already be obtained from the results of the investigation of the stability of a tangential discontinuity. Thus, for example, it is obvious that a longitudinal magnetic field which ensures the stability of a tangential discontinuity will also stabilize any other, smoother velocity profile with the same maximum change in velocity. Therefore any plane-parallel

flow along a constant magnetic field will be stable if condition (6.26) is satisfied, in which, as \(v_0\), one should take the maximum velocity difference in the flow.

The results given above refer to an ideal medium, whose viscosity and electrical resistance may be neglected. Theoretical and experimental investigations of the stability of plane-parallel flows of a real fluid confirm the general conclusion about the stabilizing action of the magnetic field.

In paper \({}^{112}\) it is shown that, as in ordinary hydrodynamics, for plane-parallel flows of an incompressible fluid in a longitudinal magnetic field the most “dangerous” are longitudinal disturbances, i.e. disturbances with a wave vector directed along the velocity of the fluid. Such disturbances lead to instability at the smallest Reynolds numbers.

The stability of a plane-parallel flow between two parallel planes in the presence of a longitudinal magnetic field was considered in paper \({}^{138}\) under the assumption that the magnetic Reynolds number \(R_m \ll 1\). In this paper it is shown that the critical Reynolds number \(R_{\mathrm{cr}}\), at which the flow becomes unstable, increases monotonically with increasing magnetic-field strength or, more precisely, with increasing dimensionless parameter

\[ q=\frac{H^2 d\sigma'}{\rho U_0}, \tag{6.29} \]

where \(\sigma'=\dfrac{\sigma}{c_0^2}\) is the conductivity in electromagnetic units, \(d\) is the half-width of the flow, and \(U_0\) is the velocity at the center of the flow. It is easy to verify that the dimensionless parameter \(q\) is expressed in terms of the Reynolds number \(R\) and the numbers \(Q\) introduced above (equation (6.11)) or \(M\) (equation (4.24)):

\[ q=\frac{Q}{R}=\frac{M^2}{R}. \tag{6.30} \]

For comparison with the results of other investigations, and also for the sake of uniformity, it is convenient everywhere to use, instead of the numbers \(q\) and \(Q\), the dimensionless parameter

\[ M=Hd\sqrt{\frac{\sigma}{\rho \nu c_0^2}}. \tag{6.31} \]

Fig. 8.

The dependence of \(R_{\mathrm{cr}}\) on \(M\) for the stability problem considered in \({}^{138}\) of a flow between parallel planes in the presence of a longitudinal magnetic field is shown by the dashed line in Fig. 8.

The investigation of the stability of an analogous flow in a transverse (perpendicular to the bounding planes) field was carried out in paper \({}^{98}\), also under the assumption \(R_m \ll 1\). In this case the principal effect of the magnetic field consists in changing the velocity profile; in comparison with this effect, the direct influence of the magnetic field on the stability of the flow is neglected. As was already noted in Section 4, in a transverse field, instead of a parabolic velocity profile, there is realiz—

is a profile with an almost uniform velocity at the center of the flow and a narrow transition layer near the walls. Since the transition layer has a thickness of order \(d/M\), as the magnetic-field strength increases the effective Reynolds number for the flow under consideration rapidly decreases. It is therefore natural to expect that the stability of such a flow increases with increasing field strength \(^{104}\). Calculations carried out in \(^{98}\) confirm this conclusion. In this case the stabilizing action of the magnetic field proves to be still stronger than in a longitudinal field. The dependence of \(R_{\mathrm{cr}}\) on \(M\) is shown in Fig. 8 by the solid line.

Experimental investigations of mercury flows in rectangular channels and tubes in a transverse magnetic field, carried out in \(^{75,113,114}\), indicate that a sufficiently strong magnetic field suppresses turbulence. Thus, in \(^{114}\) it was established experimentally that for \(M > R/900\) the flow becomes laminar. In these experiments it was possible to obtain laminar flow up to Reynolds-number values \(R = 10^5\). Thus, the stabilizing action of the magnetic field is also confirmed by direct experiments.

The magnetic field also stabilizes the flow of a viscous incompressible fluid between two rotating concentric cylinders. This problem was considered in \(^{29}\) for the case in which the magnetic field is directed along the axis of the cylinders and the cylinders rotate in the same direction. Under the assumption that the difference between the radii of the cylinders is small in comparison with the radii themselves, a relation was obtained between the critical Taylor number, at which the motion becomes unstable, and the dimensionless parameter defined above, \(Q = M^2\). The critical Taylor number increases rapidly with increasing parameter \(Q\). The stabilizing action of the magnetic field, according to the results of this work, is so great that in a field with an intensity of about \(10^4\) oersted it can already be detected in electrolytes.

In conclusion, we note that the influence of a magnetic field on the stability of all the flows considered of an incompressible viscous fluid possessing finite conductivity is characterized by one and the same dimensionless parameter \(M\). This parameter can be written in the form

\[ M=\frac{H}{\sqrt{4\pi\rho}}\sqrt{\frac{L^2}{\nu\nu_m}}, \tag{6,32} \]

where \(L\) is the characteristic dimension of the system, and it has the meaning of the ratio of the magnetohydrodynamic velocity \(H/\sqrt{4\pi\rho}\) to the geometric mean of the velocities \(\nu_m/L\) and \(\nu/L\), which characterize the equalization of nonuniformities of the magnetic field and of the velocity field as a result of Joule losses and viscous dissipation. The dimensionless parameter \(M\) serves as a numerical characteristic of the influence of the magnetic field on the dynamical behavior of a conducting fluid. The condition \(M \gg 1\) is the condition for strong interaction between the magnetic field and hydrodynamic motion, as follows from expression (4,24) for the velocity profile in a transverse magnetic field, and also from the results of the study of flow stability presented in the present section. Therefore, in those cases where the viscosity of the medium cannot be neglected, to characterize the relative role of magnetohydrodynamic effects one should use the parameter \(M\) instead of the dimensionless parameter \(HL/\nu_m\sqrt{4\pi\rho}\) proposed in \(^{104}\). The latter characterizes the influence of the magnetic field on those processes for which viscosity is inessential.

7. MAGNETOHYDRODYNAMIC TURBULENCE

In Section 5 it was shown that any motion of a conducting medium in which fluid points located on one and the same line of force move away from one another leads to an increase in the intensity of the magnetic field (provided, of course, that dissipation does not extinguish the field before it has time to increase). Turbulent motion, in particular, has this property, since in a turbulent medium the distance between any two fluid particles, on the average, increases with time. Therefore turbulence in a conducting medium serves as one of the possible mechanisms for the amplification of a weak initial magnetic field. This mechanism attracts special attention in connection with the origin of cosmic magnetic fields, since the conductivity of cosmic gas masses, owing to their high degree of ionization, is close to the conductivity of metallic conductors, and, in view of the enormous scales involved, the natural state of motion should precisely be turbulent motion.

Unfortunately, at the present time there is not only no rigorous theory of magnetic turbulence—such a theory is lacking even for ordinary turbulence—but there are not even reliable qualitative ideas concerning the character of turbulent motion of a conducting medium in the presence of a magnetic field. This is due not so much to the complexity of the problem, which does not lend itself to a consistent theoretical analysis, as to the practically complete absence of experimental data. As will be seen below, mercury and molten metals do not make it possible to realize the conditions necessary for the laboratory investigation of the most essential properties of magnetic turbulence. These properties, apparently, are manifested under cosmic conditions. However, the direct acquisition of data on cosmic magnetic turbulence represents a very difficult observational problem of astrophysics, still far from solution. Therefore at present, despite the great interest and the considerable number of works dealing with magnetic turbulence, there exist only rather scattered and often contradictory considerations of a qualitative nature, or investigations that amount to a formal extension of known methods of the theory of ordinary turbulence, the applicability of which to this essentially new field is in a number of cases doubtful.

Let us dwell in more detail on the principal questions of the theory of magnetic turbulence.

a) Condition for the growth of the magnetic field in a turbulent medium

The least complicated problem is that of the amplification of a weak initial magnetic field in a turbulent conducting medium (although here too, as we shall see, there are different points of view), since with a sufficiently weak magnetic field the turbulence must have the usual character. Indeed, in the equation of motion of the medium (1,22) the magnetic field enters in the form of terms quadratic in \(H\), whose influence for small \(H\) is negligibly small*).

The properties of ordinary turbulence at large Reynolds numbers are well known. Motion on scales of turbulence much smaller than the principal one, denoted below by \(l\), is isotropic and its prop—

* Strictly speaking, the influence of the field may be neglected only over a finite period of time, depending on the field strength and on the time interval. Over longer intervals of time the influence even of a weak magnetic field may prove substantial. This is important to bear in mind when considering stationary processes.

are completely determined by the magnitude of the energy dissipation \(\varepsilon\) per unit time per unit mass of the medium. Viscosity is significant only on turbulence scales close to the smallest, “internal,” scale \(\lambda_0\). The small number of parameters determining the properties of turbulence on scales smaller than the principal one makes it possible to use similarity considerations based on the dimensions of the quantities under consideration. Thus, the relative velocity of two points separated by a distance \(\lambda\) \((\lambda_0 \ll \lambda \ll l)\) is, in order of magnitude, determined by the expression

\[ v_\lambda \simeq (\varepsilon \lambda)^{1/3}. \tag{7,1} \]

In concrete cases the constant \(\varepsilon\) can, with accuracy sufficient for practical purposes, be expressed in terms of the characteristics of the outer scale of turbulence,

\[ \varepsilon \simeq \frac{v^3}{l}, \tag{7,2} \]

where \(v\) is the change in velocity over the extent of the largest scale of turbulence \(l\). The internal scale of turbulence and the velocity \(v_0\) characteristic of this scale are determined, in order of magnitude, by the expressions

\[ \lambda_0 \simeq \left(\frac{\nu^3}{\varepsilon}\right)^{1/4} \quad \text{and} \quad v_0 \simeq (\varepsilon \nu)^{1/4}. \tag{7,3} \]

Let a weak magnetic field have somehow appeared in a turbulent conducting fluid. There are a number of mechanisms (which we shall not dwell on) that make it possible to explain the occurrence of such a field due to currents in an unevenly heated or inhomogeneously moving medium.\(^{130}\) We are interested in the further fate of this field. We shall regard the fluid as incompressible. The behavior of the magnetic field is determined by the induction equation (1,20), which, in the case of an incompressible medium, after scalar multiplication of both sides by \(\mathbf H\), can be written in the form (cf. (5,4))

\[ \frac{1}{2}\frac{dH^2}{dt} = \mathbf H(\mathbf H \nabla)\mathbf v + \nu_m \mathbf H \Delta \mathbf H. \tag{7,4} \]

As was already indicated above, the induction effects expressed by the first term on the right-hand side, which alone can lead to an increase in the energy of the magnetic field, play an essential role only under the condition \(R_m > 1\). Otherwise the main role is played by the dissipative term of equation (7,4), and induction amplification of the field is impossible. Hence it follows at once that a magnetic field characterized by the scale \(\lambda\) will be amplified by turbulence only under the condition

\[ R_m(\lambda) = \frac{v_\lambda \lambda}{\nu_m} > 1, \tag{7,5} \]

where \(v_\lambda\) is the turbulent velocity of eddies of scale \(\lambda\). Using expressions (7,1)—(7,3), we obtain the following criterion for the growth of a field whose scale of inhomogeneities is equal to \(\lambda\):

\[ \lambda > \lambda_0 \left(\frac{\nu_m}{\nu}\right)^{3/4} = l/R_m^{3/4}. \tag{7,6} \]

Here \(R_m = \dfrac{vl}{\nu_m}\) is the magnetic Reynolds number of the turbulent motion as a whole. Since \(\lambda\) is in any case no greater than the outer scale of turbulence \(l\), a necessary condition for amplification of the field by turbulence is the condition \(R_m > 1\). In laboratory experiments with mercury

the number \(R_m\) was of order \(10^{-2}\), and with liquid sodium \(\lesssim 1\). Under these conditions dissipation suppresses induction effects, and amplification of a weak field cannot be observed.

Let us estimate the rate of growth of the magnetic energy, assuming that \(R_m(\lambda) \gg 1\) down to the smallest scales of turbulence (this corresponds to the conditions in interstellar gas) and that the density of magnetic energy is small in comparison with the density of kinetic energy of the smallest scales of turbulence, i.e. \(H^2 \ll \rho(\varepsilon \nu)^{1/2}\). This makes it possible to neglect dissipation of the field and the back reaction of the magnetic field on the turbulent motion. Under these assumptions, from equation (7.4), after averaging, we find

\[ \frac{1}{2\overline{H^2}} \frac{d\overline{H^2}}{dt} = \frac{\overline{H^2 \partial v_H/\partial x_H}}{\overline{H^2}} \simeq \overline{\frac{\partial v_H}{\partial x_H}} \simeq \left(\frac{v_\lambda}{\lambda}\right), \tag{7.7} \]

where \(\partial v_H/\partial x_H\) denotes the derivative in the direction of the field of the velocity component in the same direction. It is easy to see that this estimate corresponds to the idea of stretching of magnetic lines of force by a turbulently moving medium, since the quantity on the right-hand side expresses the mean rate of elongation of an arbitrary line attached to the medium. The principal contribution to this quantity is made by the smallest scales of turbulence. Therefore, taking (7.3) into account, we have

\[ \left(\frac{v_\lambda}{\lambda}\right) \simeq \frac{v_0}{\lambda_0} \simeq \left(\frac{\varepsilon}{\nu}\right)^{1/2}. \]

Thus, the initial magnetic field, so long as its energy is still small, grows exponentially with time with a coefficient of order \((\varepsilon/\nu)^{1/2}\) in the exponent:

\[ \overline{H^2}=\overline{H_0^2} e^{\alpha\left(\frac{\varepsilon}{\nu}\right)^{1/2} t}. \tag{7.8} \]

Here \(\alpha\) is a dimensionless constant of order unity. The exponential character of the growth of the field follows directly from the linearity with respect to \(\mathbf{H}\) of the induction equation under the assumption that the distribution of the turbulent velocity is independent of the field strength.

The fastest increase occurs in that part of the magnetic energy which is associated with the smallest scales of the magnetic field, since, owing to the properties of turbulence, \(v_\lambda/\lambda\) is maximal for these scales. At the same time, however, the field also grows on larger scales, although more slowly \(^{130,120}\). The growth rate on a scale \(\lambda \gg \lambda_0\) is determined, according to (7.7), by the ratio \(v_\lambda/\lambda\) for this scale. By virtue of (7.1) this ratio is equal to \((\varepsilon/\lambda^2)^{1/3}\), i.e. it is inversely proportional to the size of the homogeneous part of the field to the power \(2/3\). In particular, in the case when \(\nu_m>\nu\) and condition (7.6) is not satisfied for the smallest scales, field growth is possible only on those scales for which \(R_m(\lambda)>1\), and its rate is determined by the ratio \(v_\lambda/\lambda\) for the smallest of these scales.

The growth of the field in a turbulent conducting medium was first considered by Batchelor \(^{12}\). Since his conclusions differ substantially in a number of points from those given above, we shall discuss them in more detail. Batchelor starts from the analogy between the induction equation and the equation of ordinary hydrodynamics for the vorticity \(\boldsymbol{\omega}=\frac{1}{2}\operatorname{rot}\mathbf{v}\). The latter, similarly to equation (7.4), can also be brought to the form (for an incompressible fluid)

\[ \frac{1}{2}\frac{d\omega^2}{dt} = \boldsymbol{\omega}(\boldsymbol{\omega}\nabla)\mathbf{v} + \nu\boldsymbol{\omega}\Delta\boldsymbol{\omega}. \tag{7.9} \]

In the case of homogeneous turbulence, from this follows the equation for the mean-

... of its squared vorticity

\[ \frac{1}{2}\frac{d\overline{\omega^{2}}}{dt} = \omega^{2}\frac{\overline{\partial v_{\omega}}}{\partial x_{\omega}} - \nu \overline{(\nabla \omega_i)^2}. \tag{7.10} \]

The first term on the right-hand side of equation (7.10) expresses the usual mechanism of vorticity growth in turbulence due to the stretching of vortex tubes, coinciding with the mechanism of growth of magnetic-field intensity when its lines of force are stretched. Experience shows that in ordinary turbulence one always has
\[ \omega^2\frac{\overline{\partial v_{\omega}}}{\partial x_{\omega}}>0. \]
Moreover, for large \(R\) both terms on the right-hand side of equation (7.10) are large compared with their difference, i.e. the decay of vortices under the action of viscosity is approximately compensated by their growth as a result of the stretching of vortex filaments.

An equation for the mean square of the field intensity, analogous to equation (7.10), can be obtained from (7.4) on the assumption that, as a result of turbulent mixing, the magnetic field rapidly becomes statistically homogeneously distributed in space (is a stationary random function of the coordinates):

\[ \frac{1}{2}\frac{d\overline{H^{2}}}{dt} = H^{2}\frac{\overline{\partial v_H}}{\partial x_H} - \nu_m \overline{(\nabla H_i)^2}. \tag{7.11} \]

Batchelor makes the following essential assumption: the statistical distributions of \(\omega\) and \(\mathbf H\) (apart from their absolute values) are identical, or become so rapidly compared with the time of appreciable change of the mean square of the field. Under this assumption, comparison of equations (7.10) and (7.11) implies that in stationary turbulence the two terms on the right-hand side of equation (7.11) compensate each other if \(\nu_m=\nu\)*). For \(\nu_m>\nu\) dissipation predominates and the field decays. For \(\nu_m<\nu\) inductive amplification of the field predominates. Thus, according to Batchelor, the condition for field growth in turbulent motion is

\[ \nu_m<\nu. \tag{7.12} \]

This condition differs from the condition (7.6) given above. It coincides with the latter only in the case when the scale of the inhomogeneities of the magnetic field coincides with the inner scale of the turbulence, or, in other words, when the mean field on scales larger than \(\lambda_0\) is negligibly small. In the general case Batchelor’s criterion, containing no characteristic length of the field inhomogeneities, is inapplicable. Indeed, at large scales the influence of Joule dissipation is smaller, and the field will grow under a condition (7.6) less restrictive than (7.12).

Batchelor regards condition (7.12) as valid for an arbitrary initial field. His argument in favor of the similarity of the statistical properties of \(\mathbf H\) and \(\omega\) is based on the assumption that the process of stretching the magnetic-field lines of force transfers magnetic energy from large-scale pulsations of \(\mathbf H\) to small-scale ones, thereby bringing the distribution of \(\mathbf H\) closer to the distribution of \(\omega\). Regarding this assumption

*) With the corresponding boundary conditions for the turbulent velocity and magnetic field, this would mean stationarity of both the turbulence and the magnetic field. It should be noted, however, that in a stationary state one cannot neglect the back reaction of the magnetic field on the turbulence (see the preceding note). Therefore the behavior of \(\mathbf H\) and \(\omega\) ultimately does not coincide even in the most favorable case, when the initial magnetic field is everywhere proportional to the vorticity and \(\nu\) and \(\nu_m\).

it should be noted that, in view of the linearity of the induction equation with respect to \(\mathbf H\), there is no direct transfer of magnetic energy from larger scales to smaller ones. The energy of the magnetic field associated with small scales grows not at the expense of the energy of the large-scale field, but at the expense of the kinetic energy of the small-scale motions of the medium. On the other hand, the energy of the large-scale inhomogeneities of the field can decrease only through dissipation or as a result of the conversion of magnetic energy into kinetic energy of the medium. The latter mechanism makes possible an indirect transfer of magnetic energy from large scales to smaller ones through the kinetic energy of the medium; however, in the case under consideration such a transfer is impossible, if only because the field is assumed to be weak and cannot affect the motion of the medium. In this respect the behavior of the magnetic field in turbulent motion differs essentially from the behavior of vorticity. Thus, the rapid approach of the field distribution to the vorticity distribution assumed by Batchelor in fact does not occur.

Formally, this difference is also manifested in the incompleteness of the analogy between the equations for vorticity and for the magnetic-field intensity. The analogy would be sufficiently deep if the dependence of vorticity on velocity were determined only by equation (7.9). In reality this is not so. The vorticity distribution is directly kinematically connected with the velocity distribution by the relation \(\boldsymbol\omega=\frac{1}{2}\operatorname{rot}\mathbf v\). This leads to the actual nonlinearity of equation (7.9) for the vorticity and to the process, associated with this nonlinearity, of transfer of vorticity from large scales to smaller ones, which is the most essential property of ordinary turbulence. For the magnetic field, such an additional connection with the velocity field is obviously absent, and with it there is also absent a direct transfer of magnetic energy from some scales to others. Thus, the behavior of a disordered, chaotic magnetic field and the behavior of turbulent velocity differ essentially from one another, a fact often not taken into account in attempts to construct a theory of magnetic turbulence.

b) The problem of the stationary state

An important question in the theory of magnetic turbulence is the problem of the stationary state, i.e., the question of the extent to which the increase of magnetic energy considered above continues if the external causes producing the turbulence act unchanged in time. Up to the present this question has not been solved.^13 There exist two essentially different points of view, each of which encounters serious objections.

The assumption put forward by Batchelor^12,13 is that the growth of the energy of the magnetic field continues until the mean density of magnetic energy becomes comparable with the density of kinetic energy for the smallest scales of turbulence. Since, by virtue of (7.3), the latter is equal to \(\rho(\varepsilon\nu)^{1/2}\), according to Batchelor in the stationary state

\[ \frac{H^2}{8\pi}\simeq \rho(\varepsilon\nu)^{1/2}\simeq \frac{\rho v^2}{2}R^{-1/2}. \tag{7.13} \]

This state is reached in a time whose order of magnitude is determined from expression (7.8):

\[ \tau\simeq \left(\frac{\nu}{\varepsilon}\right)^{1/2}\ln\frac{8\pi\rho(\varepsilon\nu)^{1/2}}{H_0^2}. \tag{7.14} \]

Batchelor’s hypothesis is based on the formal analogy, considered above, between the equations for vorticity and for the magnetic-field intensity and supposes—

of the assumption concerning the identity of the statistical properties of \(\boldsymbol{\omega}\) and \(\mathbf H\). As was already indicated above, the analogy between vorticity and field strength is not complete, and the assumption that their statistical properties coincide is not justified upon closer examination. In fact, the matter reduces to the fact that Batchelor’s assumption ignores the growth of the field on turbulence scales larger than the inner scale. Since the kinetic energy associated with motions on these scales is greater than that expressed by relation (7.13), there is no apparent reason preventing the mean magnetic energy from growing to values larger than those in (7.13).

At present, the more widespread hypothesis is that, in the stationary state, the energy in magnetic turbulence is uniformly distributed between its magnetic and kinetic parts (see, for example, \(^{61,120,57}\)). Since the kinetic energy of turbulence is associated mainly with the largest scales of turbulence and is, in order of magnitude, equal to \(\rho v^2/2\), where \(v\) is the fluctuating velocity, according to this hypothesis, in the stationary state

\[ \frac{\overline{H^2}}{8\pi} \sim \frac{\rho v^2}{2}. \tag{7.15} \]

The ratio of the estimates (7.13) and (7.15) is, in order of magnitude,

\[ \frac{v^2}{(\varepsilon \nu)^{1/2}} \simeq \left(\frac{vl}{\nu}\right)^{1/2} \simeq R^{1/2} \tag{7.16} \]

and for interstellar gas, where \(R \simeq 10^9\), is very large.

This hypothesis also encounters serious difficulties. Indeed, the energy of a magnetic field satisfying condition (7.15) greatly exceeds the kinetic energy of small-scale fluctuations. This means that the Maxwell stresses of the magnetic field considerably exceed the hydrodynamic forces that produce motion on the small scales of turbulence and, consequently, completely determine the motion on these scales. If we turn to the equation of motion of the medium (1.22), then from condition (7.15) it follows that the stresses of the magnetic field, on the average, compensate the action of the nonlinear inertial term \((\mathbf v\nabla)\mathbf v\), with which the process—fundamental for turbulent motion—of transfer of kinetic energy from larger scales of motion to smaller ones is associated. Thus, in the case when the energy of the magnetic field is comparable with the kinetic energy of the fluid, it is no longer possible to speak of turbulent motion in the usual sense. The magnetic field suppresses motion on small scales and, consequently, disrupts the normal process of energy dissipation in stationary turbulence. This means that, with a constant influx of energy from outside, some other mechanism of energy dissipation is necessary in order for a stationary state to be possible.

Thus, if the hypothesis of equality of the magnetic and kinetic energies corresponds to reality, then we inevitably arrive at the conclusion that the magnetic field substantially modifies turbulent motion, if it does not suppress it completely.

This conclusion also follows from the investigations, considered in Section 6, of the stability of the motion of a conducting medium in the presence of a magnetic field. Questions of stability have a direct bearing on the problem of magnetic turbulence, since it is known that turbulent motion develops as a result of the instability of one or another laminar flow. In addition, the mechanism of energy transfer, fundamental for turbulent motion, from motions of large scales to motions of smaller scales may also be interpreted as the instability of large-scale motions, leading to the emergence

motions of small scales, since formally both the mechanism of energy transfer into the small-scale part of the turbulence spectrum and the instability are due to one and the same nonlinear term of the Navier—Stokes equation \((\mathbf{v}\nabla)\mathbf{v}\).

Investigation of the stability of the tangential discontinuity and of the other flows considered above shows that the magnetic field stabilizes the motion of a conducting medium and, consequently, hampers the development of turbulence. In paper \(^{84}\) the opposite assertion is made, namely that the magnetic field should facilitate the development of turbulence\({}^{*}\), since in magnetohydrodynamics the theorem on the conservation of the circulation of velocity (cf. equation (1.51)) is not fulfilled; this is based on a misunderstanding. The theorem on the conservation of vorticity does not make it possible to judge stability: in ordinary hydrodynamics fulfillment of this theorem does not ensure the stability of particular flows. On the other hand, a direct analysis shows that in all the cases considered the magnetic field only increases the stability of the motion.

In accordance with condition (6.26), a longitudinal magnetic field stabilizes a tangential discontinuity if the density of its energy is, in order of magnitude, comparable with or greater than the density of the kinetic energy of the medium. Since smoothing of the velocity profile can obviously only increase stability, this estimate also remains valid for velocity profiles smoother than a tangential discontinuity. The same estimate follows directly from qualitative energy considerations. Since the characteristic velocity of pulsations of scale \(\lambda\) is expressed by relation (7.1), one may assert that the magnetic field suppresses turbulence or, at least, makes it essentially anisotropic at all scales satisfying the condition

\[ \frac{\rho v_\lambda^{2}}{2}\simeq \rho(\varepsilon\lambda)^{2/3}\lessgtr \frac{\overline{H}^{2}}{8\pi}. \tag{7.17} \]

These considerations are in good agreement with the available experiments on the flow of mercury in tubes and rectangular channels in a transverse magnetic field. Experiments carried out in papers \(^{75}\) and \(^{113,114}\) show that, in the turbulent regime, increasing the transverse magnetic field leads at first to a decrease of the pressure gradient necessary to maintain a constant flow rate of the liquid. Then the pressure gradient grows in proportion to \(M/R\), where \(M\) is the dimensionless parameter (6.32), in good agreement with the theoretical result obtained in paper \(^{74}\) for laminar flow in a transverse field. This indicates suppression of turbulence and a transition from the turbulent law of resistance to the laminar one. According to \(^{114}\), laminar flow sets in for \(900M>R\), and it proved possible to stabilize the flow up to the maximum values \(R=10^{5}\) attained in the experiment. Since the transition from turbulent motion to laminar motion for all the values of \(R\) used is determined only by the ratio \(M/R\) and does not depend on \(R\) directly, it is natural to suppose that this transition is determined by a dimensionless parameter independent of the characteristics of the liquid \(\nu\) and \(\nu_m\). Such a parameter is

\[ Q=\frac{M}{R\sqrt{\nu/\nu_m}}=\frac{H}{v\sqrt{4\pi\rho}}. \tag{7.18} \]

It is easy to see that \(Q^2\) expresses the ratio of the magnetic energy of the system

\({}^{*}\) An analogous assumption is made in paper \(^{85a}\).

to its kinetic energy. Since, according to \(^{114}\), the transition to laminar flow occurs for mercury at \(M/R > 1/900\), and for mercury \(\nu/\nu_m = 1.5\cdot 10^{-7}\), the critical value of the parameter \(Q\) corresponding to complete suppression of turbulence in flow in a transverse field is

\[ Q_{\mathrm{cr}}=1/900\sqrt{1.5\cdot 10^{-7}}\simeq 3. \]

This value agrees, to within one order of magnitude, with the estimate (7.17) given above for the stabilizing value of the magnetic field. The agreement should be regarded as satisfactory, since, first, in contrast to a tangential discontinuity, the question here concerns a transverse field and, second, the value obtained for the parameter \(Q\) corresponds to complete suppression of turbulence, which apparently should occur under more stringent conditions than the onset of turbulence as a result of instability. Thus, both the theoretical consideration and the results of the available experiments lead to the conclusion that under the condition \(Q > 1\) magnetic turbulence is impossible.

c) Two-dimensional turbulence

In connection with the problem of amplification of the magnetic field in a turbulent medium, works \(^{139,132}\) are of interest, in which the behavior of the magnetic field in an incompressible conducting fluid under two-dimensional turbulent motion is considered. Generally speaking, it is not obvious that the usual ideas about turbulence can also be preserved in the case of two-dimensional motion. It is important, however, that the problem of amplification of the magnetic field in two-dimensional motion can be considered in the general form of an arbitrary velocity field, and a number of rigorous results can be obtained.

Let us dwell in more detail on plane two-dimensional motions, in which the velocity does not depend on one of the coordinates, say \(z\), and is parallel to the plane \((x,y)\) \(^{132}\). First of all, it is easy to verify that the amplification mechanism does not act on the component of the magnetic field \(H_z\) perpendicular to the plane of motion. This component can only decrease as a result of dissipation due to the finite conductivity of the medium. Indeed, in the case of two-dimensional motion, for the field component \(H_z\), from equation (5.1) we find

\[ \frac{\partial H_z}{\partial t}+\mathbf{v}\nabla H_z=\nu_m\Delta H_z. \tag{7.19} \]

This equation is analogous to the equation of heat conduction in a moving medium. It follows from it that, in the absence of an external field \(H_z\) at the boundaries of the two-dimensional region under consideration, \(H_z\) disappears with time throughout the entire space. Thus, \(H_z\) is of no further interest and may be taken equal to zero.

Using the relation \(\mathbf{H}=\operatorname{rot}\mathbf{A}\), we transform the induction equation (5.1) into an equation for the vector potential of the magnetic field \(\mathbf{A}\):

\[ \frac{\partial \mathbf{A}}{\partial t}=[\mathbf{v}\operatorname{rot}\mathbf{A}]+\nu_m\Delta\mathbf{A}. \tag{7.20} \]

The gradient of an arbitrary function, which appears in passing from equation (5.1) to equation (7.14), can be eliminated by a corresponding gauge transformation. For the two-dimensional magnetic field of interest to us \((H_z=0)\), the vector potential may be chosen in the form \(\mathbf{A}=(0,0,A)\), so that

\[ \mathbf{H}=\left(\frac{\partial A}{\partial y},\ -\frac{\partial A}{\partial x},\ 0\right),\qquad H^2=(\nabla A)^2. \tag{7.21} \]

By virtue of equation (7.20), \(A\) satisfies the following equation:

\[ \frac{\partial A}{\partial t}+(\mathbf{v}\nabla)A=\nu_m \Delta A . \tag{7.22} \]

This equation, just like equation (7.19) for \(H_z\), implies a decrease of \(A^2\), if \(A=0\) on the boundaries of the region (external fields are absent). Indeed, multiplying equation (7.22) by \(A\), integrating over all space, and omitting, under the condition of the absence of external fields, the surface integrals, we find

\[ \frac{1}{2}\frac{d}{dt}\int A^2\,dV = -\nu_m\int(\nabla A)^2\,dV = -\nu_m\int H^2\,dV . \tag{7.23} \]

It follows from this that \(A^2\) decreases monotonically until the magnetic field becomes equal to zero throughout all space. This means that continuous amplification of the field, or a process stationary on the average, is impossible for any two-dimensional motion. This result is essentially another formulation of the proof given in Section 5 of the impossibility of a two-dimensional stationary dynamo.

The monotonic decrease of \(A^2\) still does not mean that \(H^2\) must also decrease monotonically. Indeed, since \(H^2=(\nabla A)^2\), the magnetic field may grow despite the decrease of \(A\), if, as a result of strong mixing, the gradients of \(A\) grow. However, the growth of the gradients of \(A\) is limited by dissipation and ultimately \((\nabla A)^2\) will decrease together with \(A\). Thus, field growth is possible only in the initial stage of the process, provided that the initial field is sufficiently homogeneous. The maximum value of the field and the time during which this value is reached may be estimated as follows\(^{132}\). Let the initial field be characterized by a scale \(\lambda\), larger than the external scale of the turbulence \(l\), and by the amplitude \(A_0\) of the vector potential

\[ H_0 \simeq \frac{A_0}{\lambda}. \]

The time of macroscopic equalization of the field as a result of turbulent mixing can be expressed in terms of the turbulent viscosity \(\nu_{\text{turb}}\simeq lv\), where \(v\) is the pulsational velocity:

\[ \tau \simeq \frac{\lambda^2}{\nu_{\text{turb}}}\simeq \frac{\lambda^2}{lv}. \tag{7.24} \]

After this time has elapsed, \(H^2=(\nabla A)^2\) will only decrease. Therefore, from equation (7.17),

\[ \overline{H^2} \simeq \frac{1}{\nu_m}\frac{\overline{A^2}}{\tau} \simeq \frac{1}{\nu_m}A_0^2\frac{lv}{\lambda^2} \simeq \frac{lv}{\nu_m}\left(\frac{A_0}{\lambda}\right)^2 \]

or

\[ \sqrt{\overline{H^2}}\simeq \sqrt{R_m}\,H_0, \tag{7.25} \]

where

\[ R_m=\frac{vl}{\nu_m} \]

is the magnetic turbulence number. Thus, the growth of the field in two-dimensional turbulence continues only for a time \(\tau\) and only under the condition \(R_m>1\), which coincides with the necessary condition for amplification of a magnetic field in a turbulent medium obtained above.

An analogous result was obtained in paper \(^{139}\), in which magnetic turbulence is considered under the assumption of cylindrical sym-

metry of the problem: the field and the turbulent velocity lie in planes passing through the axis of symmetry. The results of this work also pertain only to two-dimensional motion, although this is not stipulated in the paper. Indeed, the author proceeds from the assumption that the growth of the field as a result of the entanglement of lines of force is inevitably associated with a decrease in the scale of the field inhomogeneities and, consequently, with an increase in dissipation, which ultimately limits the growth of the field. The process considered in this work is shown schematically in Fig. 9, where \(l\) is the initial scale of the field, and \(d\) is the scale of its inhomogeneities as a result of the entanglement of lines of force. The author's arguments are valid only in the case of two-dimensional turbulence, since in three-dimensional motion the growth of the field can occur without a decrease in its scales and, consequently, without

Fig. 9.          Fig. 10.

an increase in the relative role of dissipation, as is readily seen from Fig. 10. At the same time, for real three-dimensional turbulence the introduction of “turbulent conductivity,” which according to \(^{139,48}\) should lead to more rapid dissipation of the magnetic field in a turbulent medium, is also unconvincing. Thus, the assertion that the growth of the field in turbulent motion is limited by purely dissipative processes, which is valid for two-dimensional motion, does not extend to real three-dimensional turbulence. Here the same situation is found as in the relation between two-dimensional and three-dimensional dynamos. Naturally, the results obtained under the assumption of high symmetry of the magnetic field and of the velocity field are not applicable to a chaotic three-dimensional turbulent magnetic field.

g) Application of the Methods of the Theory of Homogeneous Isotropic Turbulence in Magnetohydrodynamics

In addition to the semi-quantitative treatment of magnetic turbulence set forth above, there is a considerable number of works in which the methods of the ordinary theory of turbulence are applied to magnetic turbulence. These include, above all, the works \(^{24,25,104}\), in which equations were obtained for tensors of simultaneous correlations of the values of the velocity and the magnetic-field intensity at two points of an incompressible fluid, under the assumption that the magnetic turbulence is homogeneous and isotropic. An analogous method is applied in works \(^{86,87}\) to isotropic turbulence in a compressible medium. As in the ordinary theory of turbulence, this method does not make it possible to obtain a closed system of equations, since the number of unknown correlations grows faster than the number of equations obtained for these correlations from the equations of magnetohydrodynamics. Here this difficulty is expressed even more sharply than in ordinary hydrodynamics. A solution can be obtained only for

last stage of the degeneration of turbulence, when the Reynolds numbers are small and the nonlinear terms in the equations of motion may be neglected104, 88, 97. The final stage of degeneration of magnetic turbulence in the presence of a uniform external magnetic field was considered in work97 in the linear approximation. It is shown that the turbulence possesses a sharply expressed anisotropy: harmonics of the turbulence with wave vector along the external field decay considerably faster than the others. This work also considers the influence of the Coriolis force in the case where magnetic turbulence occurs in a rotating system of coordinates.

In work89 the generation of sound waves by isotropic magnetic turbulence is considered, as well as the excitation of magnetosonic and magnetohydrodynamic waves by turbulence in the presence of a constant external magnetic field. The analysis is carried out under the assumption that fourth-order correlations are expressed through second-order correlations in accordance with the normal law, while for second-order correlations a simple Gaussian dependence on distance is assumed. Under these assumptions it is found that in magnetic turbulence the generation of sound waves increases substantially in comparison with ordinary turbulence. The aim of the work is to show the possibility of explaining the heating of the solar corona by sound perturbations generated by turbulence in the convective zone.

Chandrasekhar36 succeeded in obtaining a closed system of equations for the correlation tensors of isotropic magnetic turbulence, calculated as averages of products of the values of the velocity or magnetic-field strength at two different points at different instants of time. The use of two-time correlations and the assumption, adopted by Chandrasekhar, that fourth-order correlations are expressed through second-order correlations in accordance with the normal law make it possible to overcome the difficulty mentioned above, connected with the incompleteness of the system. However, this solution of the problem is in a certain sense apparent, since the initial conditions for the problem of determining correlations at different instants of time are the unknown simultaneous correlations.

A second method of the theory of turbulence, which was applied in works81, 82 and later in work37 to isotropic magnetic turbulence, consists in constructing an equation for the spectral density of kinetic and magnetic energy by approximately taking account of nonlinear effects by means of certain phenomenological terms, chosen intuitively while satisfying the necessary dimensional requirements. Even in the ordinary theory of turbulence this method, developed independently by Obukhov and Heisenberg, leads to a substantial discrepancy with experiment. In magnetic turbulence the arbitrariness in the choice of nonlinear terms is still greater, owing to the presence of many parameters of the same dimensionality. In particular, in works82 and37 different expressions for the nonlinear terms are postulated. The results of these works are also substantially different. The arbitrariness in the choice of one or another form of the nonlinear terms in the equations for the spectral functions deprives this method of persuasiveness, especially since at present there is no possibility of experimentally verifying one or another consequence of the theory.

With regard to the works considered above, in which the methods of the ordinary theory of turbulence are applied to magnetic turbulence, the following general remark must be made. These works are substantially based on the assumption that there is no fundamental difference between ordinary and magnetic turbulence. In particular, it is assumed—

It appears that the natural state of motion at scales smaller than the principal one is a state of local isotropy. We have seen above that the magnetic field suppresses turbulence or, at least, leads to its anisotropy at all scales whose kinetic energy is small in comparison with the density of magnetic energy. This makes the analogy between ordinary and magnetic turbulence very doubtful and, apparently, requires qualitatively new concepts.

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1. The stability of static helical fields of a similar type with respect to small deformations is considered in [^103] [^127] [^52a] [^128a] [^142]. ↩